A First Course in Linear Algebra, 2017a

5.2. The Matrix of a Linear Transformation I 269
=
⎡
⎣
= A
| |
T (⃗e 1 ) ··· T (⃗e n )
| |
⎡ ⎤
x 1
⎢ ⎥
⎣ . ⎦
x n
⎤⎡
⎦⎢
⎣
⎤
x 1
⎥
. ⎦
x n
The desired matrix is obtained from constructing the i th column as T (⃗e i ). Recall that the set {⃗e 1 ,⃗e 2 ,···,⃗e n }
is called the standard basis of R n . Therefore the matrix of T is found by applying T to the standard basis.
We state this formally as the following theorem.
Theorem 5.6: Matrix of a Linear Transformation
Let T : R n ↦→ R m be a linear transformation. Then the matrix A satisfying T (⃗x)=A⃗x is given by
⎡
| |
⎤
A = ⎣ T (⃗e 1 ) ··· T (⃗e n ) ⎦
| |
where⃗e i is the i th column of I n ,andthenT (⃗e i ) is the i th column of A.
The following Corollary is an essential result.
Corollary 5.7: Matrix and Linear Transformation
A transformation T : R n → R m is a linear transformation if and only if it is a matrix transformation.
Consider the following example.
Example 5.8: The Matrix of a Linear Transformation
Suppose T is a linear transformation, T : R 3 → R 2 where
⎡ ⎤
⎡ ⎤
1 [ ] 0 [
T ⎣ 0 ⎦ 1
= , T ⎣ 1 ⎦
9
=
2
−3
0
0
⎡
]
, T ⎣
0
0
1
⎤
[
⎦ 1
=
1
]
Find the matrix A of T such that T (⃗x)=A⃗x for all⃗x.
Solution. By Theorem 5.6 we construct A as follows:
⎡
| |
⎤
A = ⎣ T (⃗e 1 ) ··· T (⃗e n ) ⎦
| |