![Page 1: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/1.jpg)
Abstract Linear Algebra II
Linear Algebra. Session 11
Dr. Marco A Roque Sol
11 / 06 / 2018
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 2: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/2.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 3: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/3.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 4: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/4.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider
the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 5: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/5.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system
of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 6: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/6.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:
x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 7: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/7.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 8: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/8.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 9: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/9.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now,
assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 10: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/10.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that
a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 11: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/11.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0)
does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 12: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/12.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact
but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 13: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/13.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem
is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 14: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/14.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate,
namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 15: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/15.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely,
there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 16: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/16.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors
inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 17: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/17.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides
(rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 18: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/18.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 19: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/19.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 20: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/20.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find
a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 21: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/21.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation
of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 22: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/22.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Least Squares Problems
Let’s consider the Overdetermined system of linear equations:x + 2y = 3
3x + 2y = 5x + y = 2.09
⇒
x + 2y = 3−4y = −4−y = −0.09
Now, assume that a solution (x0, y0) does exist in fact but thesystem is not quite accurate, namely, there may be some errors inthe right-hand sides (rounding errors for instance).
Problem
Find a good approximation of (x0, y0)
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 23: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/23.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 24: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/24.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach
is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 25: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/25.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit.
Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 26: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/26.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely,
we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 27: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/27.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for
a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 28: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/28.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y)
that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 29: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/29.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize
the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 30: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/30.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 31: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/31.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 32: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/32.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 33: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/33.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:
a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 34: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/34.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 35: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/35.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any
x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 36: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/36.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R
define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 37: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/37.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual
r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 38: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/38.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
One approach is the least squares fit. Namely, we look for a pair(x , y) that minimize the sum
(x + 2y − 3)2 + (3x + 2y − 5)2 + (x + y − 2.09)2
Least squares solution
System of linear equations:a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2
...am1x1 + am2x2 + · · ·+ amnxn = bm
For any x ∈ R define a residual r(x) = b− Ax
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 39: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/39.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 40: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/40.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution
x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 41: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/41.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x
to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 42: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/42.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system
is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 43: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/43.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one
thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 44: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/44.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)||
(or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 45: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/45.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently,
||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 46: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/46.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 47: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/47.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 48: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/48.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A
be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 49: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/49.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an
m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 50: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/50.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and
let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 51: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/51.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let
b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 52: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/52.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 53: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/53.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 54: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/54.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector
x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 55: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/55.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂
is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 56: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/56.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution
of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 57: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/57.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system
Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 58: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/58.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax
if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 59: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/59.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and only
if it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 60: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/60.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution
of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 61: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/61.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated
normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 62: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/62.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system
ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 63: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/63.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
The least squares solution x to the system is the one thatminimizes ||r(x)|| (or, equivalently, ||r(x)||2 ).
||r(x)||2 = (m∑i=1
(ai1x1 + ai2x2 + · · ·+ ainxn − bi )2
Let A be an m × n matrix and let b ∈ Rn
Theorem
A vector x̂ is a least squares solution of the system Ax if and onlyif it is a solution of the associated normal system ATAx = ATb
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 64: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/64.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 65: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/65.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 66: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/66.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax
is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 67: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/67.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector
in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 68: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/68.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A),
the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 69: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/69.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space
of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 70: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/70.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hence
the length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 71: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/71.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length
of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 72: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/72.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax
is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 73: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/73.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal
if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 74: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/74.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax
is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 75: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/75.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the
orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 76: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/76.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection
of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 77: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/77.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b
onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 78: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/78.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A)
that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 79: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/79.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is,
if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 80: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/80.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x)
is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 81: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/81.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal
to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 82: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/82.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).
We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 83: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/83.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know
that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 84: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/84.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ =
Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 85: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/85.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace
for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 86: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/86.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix.
Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 87: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/87.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A)
the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 88: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/88.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace
of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 89: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/89.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix
ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 90: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/90.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA.
Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 91: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/91.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus,
x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 92: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/92.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is
a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 93: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/93.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution
if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 94: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/94.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 95: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/95.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒
AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 96: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/96.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒
ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 97: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/97.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
proof
Ax is an arbitrary vector in R(A), the column space of A. Hencethe length of r(x) = b− Ax is minimal if Ax is the orthogonalprojection of b onto R(A) that is, if r(x) is orthogonal to R(A).We know that row space⊥ = Nullspace for any matrix. Inparticular, R(A)⊥ = N(A) the nullspace of the transpose matrix ofA. Thus, x̂ is a least squares solution if and only if
AT r(x) = 0 ⇐⇒ AT (b− Ax) = 0 ⇐⇒ ATAx = ATb �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 98: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/98.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 99: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/99.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 100: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/100.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system
ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 101: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/101.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb
is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 102: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/102.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 103: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/103.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 104: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/104.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 105: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/105.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find
the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 106: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/106.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution to
x + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 107: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/107.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 108: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/108.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation,
the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 109: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/109.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system
can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 110: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/110.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Corollary
The normal system ATAx = ATb is always consistent.
Example 11.15
Find the least squares solution tox + 2y = 3
3x + 2y = 5x + y = 2.09
SolutionIn matrix notation, the system can be written as
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 111: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/111.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 112: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/112.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 113: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/113.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 114: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/114.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 115: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/115.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and
the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 116: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/116.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 117: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/117.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 118: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/118.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒
(11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 119: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/119.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 120: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/120.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)
⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 121: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/121.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 23 21 1
(xy
)=
35
2.09
and the normal system is
(1 3 12 2 1
) 1 23 21 1
(xy
)=
(1 3 12 2 1
) 35
2.09
⇒(
11 99 9
)(xy
)=
(20.0918.09
)⇒{
x = 1y = 1.01
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 122: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/122.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 123: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/123.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16
Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 124: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/124.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find
the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 125: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/125.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function
that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 126: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/126.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is
the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 127: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/127.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit
to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 128: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/128.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 129: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/129.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 130: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/130.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 131: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/131.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 132: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/132.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 133: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/133.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 134: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/134.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.16Find the constant function that is the least squares fit to thefollowing data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c ⇒
c = 1c = 0c = 1c = 2
⇒
1012
c ⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 135: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/135.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 136: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/136.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then,
the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 137: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/137.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 138: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/138.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
)
1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 139: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/139.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 140: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/140.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =
(1 1 1 1
) 1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 141: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/141.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 142: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/142.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 14(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 143: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/143.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1
(mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 144: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/144.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 145: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/145.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus,
the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 146: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/146.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 147: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/147.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the normal system is
(1 1 1 1
) 1111
c =(
1 1 1 1)
1012
c = 1
4(1 + 0 + 1 + 2) = 1 (mean arithmetic value)
Thus, the constant function is
f (x) = 1
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 148: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/148.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 149: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/149.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 150: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/150.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find
the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 151: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/151.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function
that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 152: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/152.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is
the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 153: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/153.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit
tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 154: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/154.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 155: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/155.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 156: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/156.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 157: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/157.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 158: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/158.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 159: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/159.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 160: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/160.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 161: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/161.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.17
Find the linear polynomial function that is the least squares fit tothe following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1+c2x ⇒
c1 = 1
c1 + c2 = 0c1 + 2c2 = 1c1 + 3c2 = 2
⇒
1 01 11 21 3
(c1c2
)=
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 162: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/162.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 163: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/163.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then,
the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 164: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/164.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 165: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/165.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
)
1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 166: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/166.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 167: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/167.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 168: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/168.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
)
1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 169: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/169.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 170: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/170.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
)
(c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 171: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/171.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 172: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/172.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)
⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 173: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/173.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 174: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/174.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus,
the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 175: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/175.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 176: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/176.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Then, the nomal system is
(1 1 1 10 1 2 3
) 1 01 11 21 3
(c1c2
)=
(1 1 1 10 1 2 3
) 1012
(
4 66 14
) (c1c2
)=
(48
)⇒{
c1 = 0.4c2 = 0.4
Thus, the linear function is
f (x) = 0.4 + 0.4x
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 177: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/177.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 178: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/178.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 179: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/179.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find
the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 180: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/180.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function
that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 181: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/181.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is
the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 182: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/182.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fit
to the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 183: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/183.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 184: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/184.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 185: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/185.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 186: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/186.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 187: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/187.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
Example 11.18
Find the quadratic polynomial function that is the least squares fitto the following data
x 0 1 2 3
f(x) 1 0 1 2
Solution
f (x) = c1 + c2x + c3x2 ⇒
c1 = 1
c1 + c2 + c3 = 0c1 + 2c2 + 4c3 = 1c1 + 3c2 + 9c3 = 2
⇒
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 188: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/188.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 189: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/189.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 190: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/190.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 191: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/191.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒
Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 192: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/192.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then,
the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 193: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/193.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 194: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/194.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 195: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/195.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 196: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/196.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 197: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/197.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 198: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/198.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1012
⇒Then, the nomal system is
1 1 1 10 1 2 30 1 4 9
1 0 01 1 11 2 41 3 9
c1
c2c3
=
1 1 1 10 1 2 30 1 4 9
1012
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 199: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/199.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 200: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/200.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 201: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/201.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 202: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/202.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 203: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/203.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 204: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/204.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus,
the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 205: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/205.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 206: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/206.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Least Squares Problems.
4 6 146 14 36
14 36 98
c1c2c3
=
48
22
⇒
c1 = 0.9c2 = −1.1c3 = 0.5
Thus, the quadratic function is
f (x) = 0.9− 1.1x + 0.5x2
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 207: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/207.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 208: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/208.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 209: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/209.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let
< ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 210: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/210.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote
the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 211: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/211.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product
in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 212: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/212.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 213: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/213.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 214: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/214.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors
v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 215: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/215.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn
form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 216: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/216.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set
ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 217: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/217.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal
to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 218: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/218.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other:
< vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 219: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/219.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0
for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 220: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/220.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 221: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/221.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If,
in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 222: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/222.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition,
all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 223: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/223.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are
of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 224: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/224.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length,
vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 225: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/225.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk
iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 226: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/226.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled
an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 227: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/227.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 228: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/228.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance,
The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 229: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/229.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basis
e1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 230: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/230.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1).
Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 231: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/231.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthogonal sets
Let < ·, · > denote the scalar product in Rn
Definition
Nonzero vectors v1, v2, · · · , vk ∈ Rn form an orthogonal set ifthey are orthogonal to each other: < vi , vj >= 0 for all i 6= j .
If, in addition, all vectors are of unit length, vi , v1, v2, · · · , vk iscalled an orthonormal set.
For instance, The standard basise1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), · · · , en = (0, 0, 0, ..., 1). Itis an orthonormal set.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 232: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/232.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 233: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/233.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 234: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/234.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose
v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 235: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/235.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn
is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 236: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/236.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis
for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 237: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/237.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn
(i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 238: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/238.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and
an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 239: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/239.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 240: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/240.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 241: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/241.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let
x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 242: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/242.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and
y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 243: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/243.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvn
where xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 244: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/244.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere
xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 245: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/245.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 246: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/246.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=
∑ni=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 247: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/247.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 248: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/248.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =
√∑ni=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 249: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/249.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Orthonormal bases
Suppose v1, v2, · · · , vn is an orthonormal basis for Rn (i.e., it is abasis and an orthonormal set).
Theorem
Let x = x1v1 + x2v2 + · · ·+ xnvn and y = y1v1 + y2v2 + · · ·+ ynvnwhere xi , y1 ∈ R
i) < x, y >=∑n
i=i xiyi
i) ||x|| =√∑n
i=i xiyi
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 250: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/250.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 251: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/251.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 252: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/252.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 253: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/253.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 254: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/254.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 255: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/255.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 256: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/256.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =
n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 257: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/257.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 258: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/258.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows
from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 259: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/259.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i)
when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 260: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/260.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
proof
i)
< x, y >=
⟨n∑i=i
xivi ,n∑j=i
yjvj
⟩=
n∑i=i
xi
⟨vi ,
n∑j=i
vj
⟩=
n∑i=i
xi
n∑j=i
yj 〈vi , vj〉 =n∑i=i
xiyi
ii) follows from i) when y = x �
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 261: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/261.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 262: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/262.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V
is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 263: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/263.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace
of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 264: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/264.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn.
Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 265: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/265.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be
the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 266: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/266.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projection
of a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 267: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/267.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn
onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 268: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/268.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 269: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/269.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V
is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 270: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/270.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace
spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 271: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/271.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, then
p = <x,v><v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 272: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/272.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp =
<x,v><v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 273: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/273.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 274: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/274.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V
admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 275: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/275.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an
orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 276: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/276.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis
v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 277: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/277.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 278: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/278.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 279: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/279.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed,
< p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 280: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/280.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=
∑kj=i
<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 281: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/281.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >=
<x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 282: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/282.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 283: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/283.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒
< x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 284: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/284.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒
(x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 285: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/285.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒
(x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 286: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/286.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Suppose V is a subspace of Rn. Let p be the orthogonal projectionof a vector x ∈ Rn onto V.
If V is a one-dimensional subspace spanned by a v, thenp = <x,v>
<v,v>v
If V admits an orthogonal basis v1, v2, · · · , vk , then
p =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vk >
< vk , vk >vk
Indeed, < p, vi >=∑k
j=i<x,vj><vj ,vj>
< vj , vi >= <x,vi><vi ,vi>
< vi , vi >=
< x, vi >⇒ < x− p, vi >= 0⇒ (x− p)⊥vi ⇒ (x− p)⊥V.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 287: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/287.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 288: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/288.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 289: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/289.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 290: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/290.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If
v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 291: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/291.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn
is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 292: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/292.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis
for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 293: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/293.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn,
then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 294: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/294.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 295: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/295.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 296: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/296.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector
x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 297: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/297.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 298: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/298.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 299: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/299.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If
v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 300: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/300.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn
is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 301: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/301.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis
for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 302: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/302.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn,
then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 303: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/303.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 304: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/304.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 305: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/305.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector
x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11
![Page 306: Linear Algebra. Session 11 - Texas A&M Universityroquesol/Math_304_Fall_2018... · 2018-11-15 · Session 11. Abstract Linear Algebra II Least Squares Problems Orthogonal Sets Least](https://reader033.vdocuments.us/reader033/viewer/2022050600/5fa750b84632e33165149e32/html5/thumbnails/306.jpg)
Abstract Linear Algebra IILeast Squares ProblemsOrthogonal Sets
Orthogonal Sets.
Coordinates relative to an orthogonal basis
Theorem
If v1, v2, · · · , vn is an orthogonal basis for Rn, then
x =< x, v1 >
< v1, v1 >v1 +
< x, v2 >
< v2, v2 >v2 + ... +
< x, vn >
< vn, vn >vn
for any vector x ∈ Rn
Corollary
If v1, v2, · · · , vn is an orthonormal basis for Rn, then
z =< x, v1 > v1+ < x, v2 > v2 + ...+ < x, vn > vn
for any vector x ∈ Rn.
Dr. Marco A Roque Sol Linear Algebra. Session 11