Download Linear Transformations: Matrix of a Linear Transformation

A FIRST COURSE
IN LINEAR ALGEBRA
An Open Text by Ken Kuttler
Linear Transformations: Matrix of a Linear
Transformation
Lecture Notes by Karen Seyffarth∗
Adapted by
LYRYX SERVICE COURSE SOLUTION
∗
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Linear Transformations: Matrix of a Linear Transformation
Page 1/13
Recall: Linear Transformations
Definition
A transformation T : Rn → Rm is a linear transformation if it satisfies the
following two properties for all ~x , ~y ∈ Rn and all (scalars) a ∈ R.
1
T (~x + ~y ) = T (~x ) + T (~y )
2
T (a~x ) = aT (~x )
Linear Transformations: Matrix of a Linear Transformation
(preservation of addition)
(preservation of scalar multiplication)
Linear Transformations
Page 2/13
Matrix Transformations
Theorem
Let T : Rn → Rm be a linear transformation. Then we can find an n × m
matrix A such that
T (~x ) = A~x
In this case, we say that T is induced, or determined, by A and we write
TA (~x ) = A~x
Linear Transformations: Matrix of a Linear Transformation
Linear Transformations
Page 3/13
Problem

a+b
a
 b+c 
3
4


The transformation T : R → R defined by T b  = 
 a − c  for
c
c −b
each ~x ∈ R3 is another matrix transformation, that is, T (~x ) = A~x for
some matrix A. Can you find a matrix A that works?



Solution
First, since T : R3 → R4 , we know
consider the product


? ? ? 
 ? ? ? 


 ? ? ? 
? ? ?
that A must have size 4 × 3. Now

a+b
a
 b+c 

b =
 a − c ,
c
c −b


and try to fill in the values of the matrix.
Linear Transformations: Matrix of a Linear Transformation
Linear Transformations
Page 4/13
Solution (continued)
We can deduce from the product that T is induced by the matrix


1
1
0
 0
1
1 
.
A=
 1
0 −1 
0 −1
1
Linear Transformations: Matrix of a Linear Transformation
Linear Transformations
Page 5/13
Is there an easier way to find the matrix of T ?
For some transformations guess and check will work, but this is not an
efficient method. The next theorem gives a method for finding the matrix
of T .
Definition
The set of columns {~e1 ,~e2 , . . . ,~en } of In is called the standard basis of Rn .
Linear Transformations: Matrix of a Linear Transformation
Linear Transformations
Page 6/13
Matrix and Linear Transformations
Theorem
Let T : Rn → Rm be a linear transformation. Then T is a matrix
transformation. Furthermore, T is induced by the unique matrix
A = T (~e1 ) T (~e2 ) · · · T (~en ) ,
where ~ej is the j th column of In , and T (~ej ) is the j th column of A.
Corollary
A transformation T is a linear transformation if and only if it is a matrix
transformation.
Linear Transformations: Matrix of a Linear Transformation
Linear Transformations
Page 7/13
Problem
Let T : R2 → R2 be a linear transformation defined by
x
x + 2y
T
=
y
x −y
for each ~x ∈ R2 . Find the matrix, A, of T .
Solution
To find A, we must find T (~e1 ) and T (~e2 ), where ~e1 and ~e2 are the standard basis
vectors of R2 .
1
1 + 2(0)
1
0
0 + 2(1)
2
T
=
=
and T
=
=
0
1−0
1
1
0−1
−1
The columns T (~e1 ) and T (~e2 ) become the columns of A, so
1
2
A=
,
1 −1
and T (~x ) = A~x for every ~x ∈ R2 . Therefore A is the matrix for T .
Linear Transformations: Matrix of a Linear Transformation
Finding the Matrix of a Linear Transformation
Page 8/13
Find the Matrix of T
Problem
Sometimes T is not defined so nicely for us. Suppose T is given as
1
1
0
3
T
=
, T
=
1
2
−1
2
Find the matrix A of T .
Linear Transformations: Matrix of a Linear Transformation
Finding the Matrix of a Linear Transformation
Page 9/13
Solution (continued)
We need to write ~e1 and ~e2 as a linear combination of the vectors
provided. First, find x and y such that
1
1
0
=x
+y
0
1
−1
Once we find x and y we can compute
1
1
0
T
= xT
+ yT
0
1
−1
=x
Linear Transformations: Matrix of a Linear Transformation
1
2
+y
3
2
Finding the Matrix of a Linear Transformation
Page 10/13
Solution (continued)
Finding x and y involves solving the following system of equations.
x =1
x −y =0
The solution is x = 1, y = 1.
Hence, we can find T (~e1 ) as follows.
1
1
3
1
3
4
T
=1
+1
=
+
=
0
2
2
2
2
4
This is the first column of the matrix A. Similarly, we can find T (~e2 )
which will be the second column of A. The resulting matrix is
4 −3
A=
4 −2
Linear Transformations: Matrix of a Linear Transformation
Finding the Matrix of a Linear Transformation
Page 11/13
Determining if a Transformation is Linear
Example

2x
x
.
y
Let T : R2 → R3 be a transformation defined by T
=
y
−x + 2y
One way to show that T is a linear transformation is to show that it preserves
addition and scalar multiplication. However, now that we know that linear
transformations are matrix transformations, we can use this to our advantage.

If T were a linear transformation, then T would be induced by the matrix


2 0
1
0
T
=  0 1 .
A = T (~e1 ) T (~e2 ) = T
0
1
−1 2
Since
A
x
y

2
= 0
−1



0 2x
x
x



1
y
=
,
=T
y
y
2
−x + 2y
T is a matrix transformation, and therefore a linear transformation.
Linear Transformations: Matrix of a Linear Transformation
Finding the Matrix of a Linear Transformation
Page 12/13
Example
x
xy
=
.
y
x +y
If T were a linear transformation, then T would be induced by the matrix
1
0
0 0
A = T (~e1 ) T (~e2 ) = T
T
=
.
0
1
1 1
Let T : R2 → R2 be a transformation defined by T
However,
A
x
y
=
0
1
0
1
x
y
=
0
x +y
.
We see from this that if x = 0 or y = 0, then xy = 0, so A
x
y
=T
x
y
.
But if we take x = y = 1, then
x
1
0
x
1
1
A
=A
=
while T
=T
=
,
y
1
2
y
1
2
i.e., A
1
1
6= T
1
1
. Therefore, T in not a linear transformation.
Linear Transformations: Matrix of a Linear Transformation
Finding the Matrix of a Linear Transformation
Page 13/13