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SECTION 1.1: SYSTEMS OF LINEAR EQUATIONS THE BASICS What is a linear equation? What is a system of linear equations? What is a solution of a system? What is a solution set?
2
When are two systems equivalent? Consistent vs. Inconsistent Existence & Uniqueness (Fundamental Questions about a Linear System) Solving a linear system using the method of elimination
3
MATRIX NOTATION
What is a matrix? What is a coefficient matrix vs. an augmented matrix? How do I find the size of a matrix?
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Elementary Row Operations (pg 7)
1.) Replacement: Replace one row by the sum of itself and a multiple of another row.
2.) Interchange: Interchange two rows.
3.) Scaling: Multiply all entries in a row by a nonzero constant.
When are two matrices row equivalent? SOLVING A LINEAR SYSTEM USING MATRICES
**(We will learn the method for this later. For now just recognize that a linear system can be solved using matrices.)**
Try the following elementary row operations. Begin with the augmented matrix for the system. In the last step, turn your new augmented matrix back into a linear system.
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EXAMPLES OF HOW TO SOLVE A LINEAR SYSTEM
Using Elimination Using Matrices
System
2z2y2x3
6zyx2
9zy2x
=−−
=++
=−+
2z2y2x3
12z3y3
9zy2x
=−−
−=+−
=−+
221 RRR2 →+−
25zy8
12z3y3
9zy2x
−=+−
−=+−
=−+
331 RRR3 →+−
7z7
12z3y3
9zy2x
=−
−=+−
=−+
332 RRR3
8→+−
Solution
1z
3y
2x
−=
=
=
This system is consistent & the solution is unique.
1z
12z3y3
9zy2x
−=
−=+−
=−+
33 RR7
1→−
1z
9y3
9zy2x
−=
−=−
=−+
223 RRR3 →+−
6
1z
3y
9zy2x
−=
=
=−+
22 RR3
1→−
1z
3y
8y2x
−=
=
=+
113 RRR →+−
1z
3y
2x
−=
=
=
112 RRR2 →+−
System
5y3x
8y5x2
4y3x
−=+
−=+
−=+
Look at equations 1 and 3.
Is this possible?
Solution
This system is
inconsistent. There is no solution.
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System
14z2y3x2
2zyx3
6z3y2x
=+−
−=−+
=++
2z4y7
2zyx3
6z3y2x
=−−
−=−+
=++
331 RRR2 →+−
2z4y7
20z10y5
6z3y2x
=−−
−=−−
=++
221 RRR3 →+−
2z4y7
22z6y2
6z3y2x
=−−
−=−
=++
223 RRR →+−
2z4y7
22z6y2
28z9x
=−−
−=−
=+
112 RRR →+−
2z4y7
11z3y
28z9x
=−−
−=−
=+
22 RR2
1→
8
Solution
3z
2y
1x
=
−=
=
This system is consistent & the solution is unique.
75z25
11z3y
28z9x
−=−
−=−
=+
332 RRR7 →+
3z
11z3y
28z9x
=
−=−
=+
33 RR25
1→−
3z
2y
28z9x
=
−=
=+
223 RRR3 →+
3z
2y
1x
=
−=
=
113 RRR9 →+−
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SECTION 1.2: ROW REDUCTION AND ECHELON FORMS ROW REDUCTION & ECHELON FORMS What is a leading entry?
Echelon Form (pg 19) Reduced Echelon Form (pg 19)
1.) All nonzero rows are above any rows of
all zeros.
2.) Each leading entry of a row is in a
column to the right of the leading entry of
the row above it.
3.) All entries in a column below a leading
entry are zeros.
1.) All nonzero rows are above any rows of
all zeros.
2.) Each leading entry of a row is in a
column to the right of the leading entry of
the row above it.
3.) All entries in a column below a leading
entry are zeros.
4.) The leading entry in each nonzero row is
1.
5.) Each leading 1 is the only nonzero entry
in its column.
What is an echelon matrix or a row reduced echelon matrix?
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Which of these is an echelon matrix, a row reduced echelon matrix, or neither?
963
852
741
0000
1450
0221
0000
0000
0001
00
120
6592
100
010
001
200
500
180
256
Theorem 1: Uniqueness of the Reduced Echelon Form What is a pivot position?
0000
1450
0221
What is a pivot column? What is a pivot?
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The Row Reduction Algorithm (pg 23 & 24)
1.) Begin with the leftmost nonzero column. This is the first pivot column and the pivot position is at the top.
2.) Choose a nonzero entry in the first pivot column as a pivot. You may need to interchange
rows to get your pivot into the pivot position. 3.) Use row replacement operations to create zeros in all positions below the pivot. 4.) Cover the row containing the pivot (and any rows above it) and apply steps 1-3 to the
submatrix that remains. Repeat this process until there are no more nonzero rows to modify. Your matrix should now be in echelon form.
5.) Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If the pivot is not 1, use a scaling operation to make it a 1. Your matrix should now be in row reduced echelon form.
Forward phase vs. Backward phase
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Example 1
10072
7041
Example 2
−−
−
−−
9954
0121
8543
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**Any nonzero matrix may be row reduced into more than one matrix in echelon form, using different sequences of row operations. However, the row reduced echelon form of a matrix is unique!**
SOLUTIONS OF LINEAR SYSTEMS
−
−
−
−−
−−
−−
7
5
0
10000
04100
03061
6
5
4
213461
5186122
25261
~
The associated system is now
7x
5x4x
0x3x6x
5
43
421
=
=−
=++
.
What is a general solution? Basic variables vs. Free variables What is a parametric description?
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What does “the solution set is empty” mean? Theorem 2: Existence & Uniqueness Theorem
Using Row Reduction To Solve A Linear System (pg 29)
1.) Write the augmented matrix for the system. 2.) Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon
form. Decide whether the system is consistent. If there is no solution, stop; otherwise continue to step 3.
3.) Continue row reduction to obtain the reduced echelon form. 4.) Write the system of equations corresponding to the matrix in step 3. 5.) Rewrite each nonzero equation from step 4 so that its one basic variable is expressed
in terms of any free variables appearing in the equation.
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Example 3 Find the general solution for the following system:
13x7x7x5x
16x4x9x5x2
7xx4x2x
4321
4321
4321
=−−+
=−−+
=−−+
.
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SECTION 1.3: VECTOR EQUATIONS
VECTORS IN Rn
What is Rn? What is a vector? What is a scalar multiple of a vector? Geometric Descriptions (pg 36)
*Notice that the arrow for u2 is twice as long as the arrow for u .*
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Parallelogram Rule for Addition
ALGEBRAIC PROPERTIES OF nR (pg 39)
For all u , v , w in nR and all scalars c and d:
1.) uvvu +=+ 5.) vcuc)vu(c +=+
2.) )wv(uw)vu( ++=++ 6.) uducu)dc( +=+
3.) uu00u =+=+ 7.) u)cd()ud(c =
4.) 0u)u()u(u =+−=−+
where -u denotes u*1− . 8.) uu*1 =
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What is a linear combination?
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Example 1
Can d be written as a linear combination of a , b , & c ?
=
3
2
1
a
=
6
5
4
b
−
=
2
0
2
c
−
−
=
7
11
5
d
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*Notice that a vector equation bxaxaxa nn =+++ K2211 has the same solution set as the
linear system whose augmented matrix is [ ]baaa nK21 . In particular, b can be generated
by a linear combination of 1a , 2a , K , na if and only if the linear system whose augmented
matrix is [ ]baaa nK21 is consistent.*
What is Span }v,,v,v{ nK21 ?
Geometric Description (pg 42)
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Example 2
Let
=
4
6
1
a ,
−
=
1
4
2
b ,
−
=
5
2
1
c ,
=
9
8
0
d .
Is d in { }c,b,aSpan ? (Can d be written as a linear combination of a , b , and c ?)
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LINEAR COMBINATIONS IN APPLICATIONS
Example 3 A dietician is preparing a meal consisting of foods A, B, & C. Each ounce of food A contains 2 units of protein, 3 units of fat and 4 units of carbohydrate. Each ounce of food B contains 3 units of protein, 2 units of fat and 1 unit of carbohydrate. Each unit of food C contains 3 units of protein, 3 units of fat, and 2 units of carbohydrate. If the meal must provide exactly 25 units of protein, 24 units of fat and 21 units of carbohydrate, how many ounces of each type of food should be used?
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1.4: THE MATRIX EQUATION bxA =
If A is an mxn matrix and x is a vector in nR , then what is xA ?
Example 1
If
=
654
321A and
=
9
8
7
x then
=
=
9
8
7
654
321xA
Example 2
If
=
63
52
41
A and
=8
7x then =
=8
7
63
52
41
xA
Theorem 3
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Existence of Solutions
Example 3
Let
−
−
=
43
50
21
A and
=
3
2
1
b
b
b
b . Does bxA = have a solution for all b ?
Theorem 4 *Theorem 4 is about a coefficient matrix not an augmented matrix!*
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Computation of xA (short cut) (pg 56)
=
654
321A and
=
9
8
7
x
1.) Let’s start by looking at the first row.
=
++=
509*38*27*1
9
8
7321
2.) Now let’s look at the second row.
=
++=
1229*68*57*49
8
7
654
3.) We now have that
=
+
=
122
50
122
50xA .
Example 4
=
−−
2
5
1
406
721
Example 5
=
−
−
−5
2
41
32
96
Example 6
=
t
s
r
100
010
001
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What is the identity matrix? Theorem 5
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1.5 SOLUTIONS SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS When is a linear system called homogeneous? Trivial Solution vs. Nontrivial Solution
When does the homogeneous equation 0=xA have a nontrivial solution?
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Example 1 Determine if the following system has a nontrivial solution. Then describe the solution set.
0x2xx2
0x2x3x
0x3x2x
321
321
321
=−+
=++−
=++
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Example 2 Determine if the following system has a nontrivial solution. Then describe the solution set.
0x2xx3
0x3x2x
0xxx2
321
321
321
=+−−
=−−
=−+
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PARAMETRIC VECTOR FORM What is a parametric vector equation? SOLUTIONS OF NONHOMOGENEOUS SYSTEMS Theorem 6
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Example 3 Determine if the following system has a nontrivial solution. Then describe the solution set.
0x3x6x
3x1x8x2
4x2x5x2x6x
421
543
54321
=++
=−−
−=−−++
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Writing A Solution Set (Of a Consistent System) in Parametric Vector Form (pg 71)
1.) Row reduce the augmented matrix to reduced echelon form. 2.) Express each basic variable in terms of any free variables appearing in an equation.
3.) Write a typical solution x as a vector whose entries depend on the free variable, if any.
4.) Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters.
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1.6: LINEAR INDEPENDENCE Linearly Independent vs. Linearly Dependent
Example 1
Let
−
=
1
2
1
v1 ,
−=
1
2
1
v 2 ,
−
−
=
1
2
3
v 3 , and
=
0
0
2
v 4 . Is S={ 1v , 2v , 3v , 4v } linearly
independent or linearly dependent?
34
When are the columns of a matrix A linearly independent? When is a set of 2 vectors linearly independent/dependent? Theorem 7: Characterization of Linearly Dependent Sets Theorem 8 Theorem 9
35
Example 2 Determine whether each of the following sets is linearly independent or linearly dependent.
3
2
1
,
0
0
0
4
3
2
1
,
8
2
1
0
27
3
0
,
18
2
0
,
−
6
7
5
3
2
1
,
0
3
2
,
−
6
7
1
,
5
81
9
−
−
4
12
8
,
−
−
1
3
2
3
2
1
,
0
3
2
,
6
7
4
36
Example 3
What value(s) of h make the vectors
−=
2
3
1
v1 ,
−
−
=
6
10
3
v 2 , and
−=
h
7
5
v 3 linearly
independent/linearly dependent.
37
Example 4
Give an example of a set of vector from 3R that contains
3
2
1
, spans 3R , and is linearly
independent. Show that you example works.
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1.7: INTRODUCTION TO LINEAR TRANSFORMATIONS What is a transformation?
What are the domain, codomain, image and range of mn RR:T → ?
What is a matrix transformation? When is a transformation called linear?
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Example 1
Let
−=
10
61A ,
=2
0u ,
=0
4b ,
=2
4c and define a transformation 22 RR:T → by
xA)x(T = .
a.) Find )u(T , the image of u under the transformation T.
b.) Find an x in 2R whose image under T is b .
c.) Is there more than one x whose image under T is b ?
d.) Determine if c is in the range of T.
This transformation is called a shear transformation. See page 96 for more information.
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Example 2
Define a transformation 22 RR:T → by T(x)=Ax where
−=
10
01A . Describe
geometrically what this transformation does to the triangle whose vertices are the points (-1,4), (3,1), and (2,6).
Example 3
Define a transformation, 22 RR:T → , by T(x)=2x. Show that T is a linear transformation.
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Example 4
Let
=
4
3
2
1
a ,
−
−
=
0
5
8
1
b ,
−=
7
3
6
2
c , and
−=
30
21
35
10
d . If 64 RR:T → is a linear transformation
with ( )
=
1
6
0
4
0
2
aT , ( )
−
=
4
5
2
1
1
0
bT , and ( )
−
−
=
3
1
2
0
5
4
cT . Find ( )dT .
42
1.8 THE MATRIX OF A LINEAR TRANSFORMATION
What are ne,,e,e K21 ?
Theorem 10 What is the standard matrix for a linear transformation?
Example 1
Let 23 RR:T → be the linear transformation defined by
−
+−=
31
321
3
2
1
x7x2
xx3x2
x
x
x
T . Find
the standard matrix of this transformation.
43
*Geometric Linear Transformations of 2R (pg 112-114)*
What does it mean for a transformation to be onto? What does it mean for a transformation to be one-to-one? Theorem 11 Theorem 12
44
Example 2 Give an example of each of the following: 1.) a linear transformation that is both one-to-one and onto.
2.) a linear transformation that is neither one-to-one nor onto.
3.) a linear transformation that is onto but not one-to-one.
4.) a linear transformation that is one-to-one but not onto.