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Lesson 3The Dot Product and Matrix Multiplication
Math 20
September 24, 2007
Announcements
I Problem Set 1 is on the course web site. Due September 26.
I Problem Sessions: Sundays 6–7 (SC 221), Tuesdays 1–2 (SC116)
I My office hours: Mondays 1–2, Tuesdays 3–4, Wednesdays1–3 (SC 323)
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The price of breakfast
Remember I eat two eggs, three slices of bacon, and two slices oftoast for breakfast. Then my breakfast can be summarized by theobject
b =
232
.
Suppose eggs cost $1.39 per dozen, bacon costs $2.49 per pound,and bread costs $1.99 per loaf. Assume a pound of bacon has 16slices, as does a loaf of bread. Then the price per “unit” ofbreakfast is
p =
1.39/122.49/161.99/16
=
0.120.160.12
QuestionHow much do I pay?
![Page 3: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/3.jpg)
The price of breakfast
Remember I eat two eggs, three slices of bacon, and two slices oftoast for breakfast. Then my breakfast can be summarized by theobject
b =
232
.
Suppose eggs cost $1.39 per dozen, bacon costs $2.49 per pound,and bread costs $1.99 per loaf. Assume a pound of bacon has 16slices, as does a loaf of bread. Then the price per “unit” ofbreakfast is
p =
1.39/122.49/161.99/16
=
0.120.160.12
QuestionHow much do I pay?
![Page 4: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/4.jpg)
Answer.The answer is
(0.12)(2) + (0.16)(3) + (0.12)(2) = 0.96.
My breakfast costs 96 cents.
In terms of the vectors
p =
0.120.160.12
b =
232
what have we done? We multiplied the components and addedthem.
![Page 5: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/5.jpg)
Answer.The answer is
(0.12)(2) + (0.16)(3) + (0.12)(2) = 0.96.
My breakfast costs 96 cents.
In terms of the vectors
p =
0.120.160.12
b =
232
what have we done?
We multiplied the components and addedthem.
![Page 6: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/6.jpg)
Answer.The answer is
(0.12)(2) + (0.16)(3) + (0.12)(2) = 0.96.
My breakfast costs 96 cents.
In terms of the vectors
p =
0.120.160.12
b =
232
what have we done? We multiplied the components and addedthem.
![Page 7: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/7.jpg)
The dot product of vectors
DefinitionWe p, q be vectors in Rn. We define the dot product (or innerproduct) of p and q to be the scalar
p · q = p1q1 + p2q2 + · · ·+ pnqn.
Observations:
I The dot product of two vectors is a scalar.
I The vectors need to have the same length to multiply.
I The dot product is symmetric meaning p · q is always equalto q · p.
q·p = q1p1+q2p2+· · ·+qnpn = p1q1+p2q2+· · ·+pnqn = p·q
![Page 8: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/8.jpg)
The dot product of vectors
DefinitionWe p, q be vectors in Rn. We define the dot product (or innerproduct) of p and q to be the scalar
p · q = p1q1 + p2q2 + · · ·+ pnqn.
Observations:
I The dot product of two vectors is a scalar.
I The vectors need to have the same length to multiply.
I The dot product is symmetric meaning p · q is always equalto q · p.
q·p = q1p1+q2p2+· · ·+qnpn = p1q1+p2q2+· · ·+pnqn = p·q
![Page 9: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/9.jpg)
The dot product of vectors
DefinitionWe p, q be vectors in Rn. We define the dot product (or innerproduct) of p and q to be the scalar
p · q = p1q1 + p2q2 + · · ·+ pnqn.
Observations:
I The dot product of two vectors is a scalar.
I The vectors need to have the same length to multiply.
I The dot product is symmetric meaning p · q is always equalto q · p.
q·p = q1p1+q2p2+· · ·+qnpn = p1q1+p2q2+· · ·+pnqn = p·q
![Page 10: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/10.jpg)
The dot product of vectors
DefinitionWe p, q be vectors in Rn. We define the dot product (or innerproduct) of p and q to be the scalar
p · q = p1q1 + p2q2 + · · ·+ pnqn.
Observations:
I The dot product of two vectors is a scalar.
I The vectors need to have the same length to multiply.
I The dot product is symmetric meaning p · q is always equalto q · p.
q·p = q1p1+q2p2+· · ·+qnpn = p1q1+p2q2+· · ·+pnqn = p·q
![Page 11: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/11.jpg)
The dot product of vectors
DefinitionWe p, q be vectors in Rn. We define the dot product (or innerproduct) of p and q to be the scalar
p · q = p1q1 + p2q2 + · · ·+ pnqn.
Observations:
I The dot product of two vectors is a scalar.
I The vectors need to have the same length to multiply.
I The dot product is symmetric meaning p · q is always equalto q · p.
q·p = q1p1+q2p2+· · ·+qnpn = p1q1+p2q2+· · ·+pnqn = p·q
![Page 12: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/12.jpg)
Math 20 - September 24, 2007.GWBMonday, Sep 24, 2007
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Another Example
Example
Let v =
1−14
and w =
220
. Then
v ·w
= 1 · 2 + (−1) · 2 + 4 · 0 = 0.
So vectors can have a zero inner product without either one beingzero.
![Page 14: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/14.jpg)
Another Example
Example
Let v =
1−14
and w =
220
. Then
v ·w = 1 · 2 + (−1) · 2 + 4 · 0
= 0.
So vectors can have a zero inner product without either one beingzero.
![Page 15: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/15.jpg)
Another Example
Example
Let v =
1−14
and w =
220
. Then
v ·w = 1 · 2 + (−1) · 2 + 4 · 0 = 0.
So vectors can have a zero inner product without either one beingzero.
![Page 16: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/16.jpg)
Another Example
Example
Let v =
1−14
and w =
220
. Then
v ·w = 1 · 2 + (−1) · 2 + 4 · 0 = 0.
So vectors can have a zero inner product without either one beingzero.
![Page 17: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/17.jpg)
Dot product and Length
If v =
(ab
), then
v · v
= a2 + b2 = ‖v‖2
Sometimes useful even if our vectors aren’t really physical innature.
![Page 18: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/18.jpg)
Dot product and Length
If v =
(ab
), then
v · v = a2 + b2
= ‖v‖2
Sometimes useful even if our vectors aren’t really physical innature.
![Page 19: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/19.jpg)
Dot product and Length
If v =
(ab
), then
v · v = a2 + b2 = ‖v‖2
Sometimes useful even if our vectors aren’t really physical innature.
![Page 20: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/20.jpg)
Dot product and Length
If v =
(ab
), then
v · v = a2 + b2 = ‖v‖2
Sometimes useful even if our vectors aren’t really physical innature.
![Page 21: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/21.jpg)
Math 20 - September 24, 2007.GWBMonday, Sep 24, 2007
Page3of8
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Orthogonality
v
w
v + w
If v and w make a right angle, then
‖v‖2 + ‖w‖2 = ‖v + w‖2
On the other hand,
‖v + w‖2 = (v + w) · (v + w)FOIL= v · v + 2v ·w + w ·w= ‖v‖2 + 2v ·w + ‖w‖2
So v and w are orthogonal (perpendicular) if v ·w = 0.
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Sigma notation
p · q = p1q1 + p2q2 + · · ·+ pnqn =n∑
i=1
piqi
The symbol i is an index, a “variable” which takes all integervalues between 1 and n.
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Who else could go for some flapjacks?
Ingredient Pancakes Crepes Blintzes
Flour (cups) 112
12 1
Water (cups) 0 14 0
Milk (cups) 112
14 0
Eggs 2 2 3Oil (Tbsp) 3 2 2
The yield for each recipe is 12.
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Math 20 - September 24, 2007.GWBMonday, Sep 24, 2007
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Again, let’s look at the what we’ve done in terms of the matrix
A =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
and v =
abc
(whatever they are).
We essentially took the dot product of v with every row of A, thenformed the vectors whose components were that vector.
![Page 27: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/27.jpg)
Again, let’s look at the what we’ve done in terms of the matrix
A =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
and v =
abc
(whatever they are).
We essentially took the dot product of v with every row of A, thenformed the vectors whose components were that vector.
![Page 28: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/28.jpg)
Math 20 - September 24, 2007.GWBMonday, Sep 24, 2007
Page7of8
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The matrix-vector product
Definition
Let A = [aij ] be an m × n matrix and v =
v1
v2
. . .vn
a n-vector
(column vector). The matrix-vector product of A and v is the
vector Av =
w1
w2
. . .wm
, where
wk = ak1v1 + ak2v2 + · · ·+ aknvn =n∑
j=1
akjvj ,
the dots product of kth row of A with v.
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Discussion
Dimensional considerations?
RemarkThe matrix-vector product Av is defined only when A is m × n andv is column vector in Rn. The result is in Rm.
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Discussion
Dimensional considerations?
RemarkThe matrix-vector product Av is defined only when A is m × n andv is column vector in Rn. The result is in Rm.
![Page 32: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/32.jpg)
Example
Let
A =
2 3−1 40 3
and v =
[2−1
]
Find Av.
Solution
Av =
2 · 2 + 3 · (−1)(−1) · 2 + 4 · (−1)
0 · 2 + 3 · (−1)
=
4− 1−2− 40− 3
=
1−6−3
.
![Page 33: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/33.jpg)
Example
Let
A =
2 3−1 40 3
and v =
[2−1
]
Find Av.
Solution
Av =
2 · 2 + 3 · (−1)(−1) · 2 + 4 · (−1)
0 · 2 + 3 · (−1)
=
4− 1−2− 40− 3
=
1−6−3
.
![Page 34: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/34.jpg)
Example
Let
A =
2 3−1 40 3
and v =
[2−1
]
Find Av.
Solution
Av =
2 · 2 + 3 · (−1)(−1) · 2 + 4 · (−1)
0 · 2 + 3 · (−1)
=
4− 1−2− 40− 3
=
1−6−3
.
![Page 35: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/35.jpg)
Example
Let
A =
2 3−1 40 3
and v =
[2−1
]
Find Av.
Solution
Av =
2 · 2 + 3 · (−1)(−1) · 2 + 4 · (−1)
0 · 2 + 3 · (−1)
=
4− 1−2− 40− 3
=
1−6−3
.
![Page 36: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/36.jpg)
Matrix product redefined
Another way to look at the product of a matrix and a vector isthis: The product of A and v is a linear combination of thecolumns of A using the components of v as coefficients.
(A linearcombination is a combination of scaling and adding vectors) So if
A =
2 3−1 40 3
and v =
[2−1
]
Av = a1v1 + a2v2
=
2−10
· 2 +
343
· (−1) =
4−20
+
−3−4−3
=
1−6−3
which is the same as above.
![Page 37: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/37.jpg)
Matrix product redefined
Another way to look at the product of a matrix and a vector isthis: The product of A and v is a linear combination of thecolumns of A using the components of v as coefficients. (A linearcombination is a combination of scaling and adding vectors)
So if
A =
2 3−1 40 3
and v =
[2−1
]
Av = a1v1 + a2v2
=
2−10
· 2 +
343
· (−1) =
4−20
+
−3−4−3
=
1−6−3
which is the same as above.
![Page 38: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/38.jpg)
Matrix product redefined
Another way to look at the product of a matrix and a vector isthis: The product of A and v is a linear combination of thecolumns of A using the components of v as coefficients. (A linearcombination is a combination of scaling and adding vectors) So if
A =
2 3−1 40 3
and v =
[2−1
]
Av = a1v1 + a2v2
=
2−10
· 2 +
343
· (−1) =
4−20
+
−3−4−3
=
1−6−3
which is the same as above.
![Page 39: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/39.jpg)
Matrix product redefined
Another way to look at the product of a matrix and a vector isthis: The product of A and v is a linear combination of thecolumns of A using the components of v as coefficients. (A linearcombination is a combination of scaling and adding vectors) So if
A =
2 3−1 40 3
and v =
[2−1
]
Av = a1v1 + a2v2
=
2−10
· 2 +
343
· (−1) =
4−20
+
−3−4−3
=
1−6−3
which is the same as above.
![Page 40: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/40.jpg)
Matrix product redefined
Another way to look at the product of a matrix and a vector isthis: The product of A and v is a linear combination of thecolumns of A using the components of v as coefficients. (A linearcombination is a combination of scaling and adding vectors) So if
A =
2 3−1 40 3
and v =
[2−1
]
Av = a1v1 + a2v2
=
2−10
· 2 +
343
· (−1) =
4−20
+
−3−4−3
=
1−6−3
which is the same as above.
![Page 41: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/41.jpg)
Matrix Product
Suppose we are running HDS and we know that flat breakfast friedbatter concoction preferences change from house to house. Maybeit’s something like this:
Food Frosh Lowell Dunster Pforzheimer
Pancakes 70 60 50 40Crepes 20 30 30 30
Blintzes 10 10 20 30
Let B be the matrix above. Then we can get the house breakdownof ingredients for each class.
![Page 42: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/42.jpg)
The amount of ingredients we need for the freshman class is
Ab1
=
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 43: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/43.jpg)
The amount of ingredients we need for the freshman class is
Ab1 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 44: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/44.jpg)
The amount of ingredients we need for the freshman class is
Ab1 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 45: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/45.jpg)
The amount of ingredients we need for the freshman class is
Ab1 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2
=
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 46: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/46.jpg)
The amount of ingredients we need for the freshman class is
Ab1 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 47: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/47.jpg)
The amount of ingredients we need for the freshman class is
Ab1 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 48: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/48.jpg)
The amount of ingredients we need for the freshman class is
Ab1 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70
2010
=
125
5110210270
That for the Lowell House is
Ab2 =
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
60
3010
=
1157.5
97.5210260
,
and so on.
![Page 49: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/49.jpg)
Putting this together gives a matrix
[Ab1 Ab2 Ab3 Ab4
]=
125 115 110 105
5 7.5 7.5 7.5100 97.5 82.5 67.5210 210 220 230270 260 250 240
![Page 50: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/50.jpg)
Matrix product, defined
DefinitionLet A be an m × n matrix and B a n × p matrix. Then the matrixproduct of A and B is the m × p matrix whose jth column is Abj .In other words, the (i , j)th entry of AB is the dot product of ithrow of A and the jth column of B. In symbols
(AB)ij =n∑
k=1
aikbkj .
Example
1.5 0.5 10 0.25 0
1.5 0.25 02 2 33 2 2
70 60 50 40
20 30 30 3010 10 20 30
=
125 115 110 105
5 7.5 7.5 7.5100 97.5 82.5 67.5210 210 220 230270 260 250 240
![Page 51: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/51.jpg)
Math 20 - September 24, 2007.GWBMonday, Sep 24, 2007
Page8of8
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RemarkDimensional considerations again
: In order for A and B to bemultipliable, the number of columns of A has to be equal to thenumber of rows of B. The resulting matrix as the same number ofrows as A and the same number of columns as B.
Am×nBn×p = (AB)m×p
![Page 53: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/53.jpg)
RemarkDimensional considerations again: In order for A and B to bemultipliable, the number of columns of A has to be equal to thenumber of rows of B. The resulting matrix as the same number ofrows as A and the same number of columns as B.
Am×nBn×p = (AB)m×p
![Page 54: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/54.jpg)
RemarkDimensional considerations again: In order for A and B to bemultipliable, the number of columns of A has to be equal to thenumber of rows of B. The resulting matrix as the same number ofrows as A and the same number of columns as B.
Am×nBn×p = (AB)m×p
![Page 55: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/55.jpg)
Example
Let
A =
2 3−1 40 3
and B =
[3 −11 2
]
Find AB.
Solution
2 3−1 40 3
[3 −11 2
]=
2 · 3 + 3 · 1 2 · (−1) + 3 · 2(−1) · 3 + 4 · 1 (−1) · (−1) + 4 · 2
0 · 3 + 3 · 1 0 · (−1) + 3 · 2
=
6 + 3 −2 + 6−3 + 4 1 + 80 + 3 0 + 6
=
9 41 93 6
![Page 56: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/56.jpg)
Example
Let
A =
2 3−1 40 3
and B =
[3 −11 2
]
Find AB.
Solution
2 3−1 40 3
[3 −11 2
]=
2 · 3 + 3 · 1 2 · (−1) + 3 · 2(−1) · 3 + 4 · 1 (−1) · (−1) + 4 · 2
0 · 3 + 3 · 1 0 · (−1) + 3 · 2
=
6 + 3 −2 + 6−3 + 4 1 + 80 + 3 0 + 6
=
9 41 93 6
![Page 57: Lesson03 Dot Product And Matrix Multiplication Slides Notes](https://reader035.vdocuments.us/reader035/viewer/2022081404/558e1c761a28abb25b8b46e3/html5/thumbnails/57.jpg)
Conclusions
I The product of matrices and vectors have very usefulinterpretations in various models. That’s why they’re souseful.
I Next time we’ll make sure that certain manipulations we wantto do with these products are valid. In what ways are matrixproducts like the product of real numbers? Is it commutative?Associative? And so on.