A First Course in Linear Algebra, 2017a

5.2. The Matrix of a Linear Transformation I 271
[ 4
Therefore
4
]
is the first column of A.
Computing the second column is done in the same way, and is left as an exercise.
The resulting matrix A is given by [ ] 4 −3
A =
4 −2
♠
This example illustrates a very long procedure for finding the matrix of A. While this method is reliable
and will always result in the correct matrix A, the following procedure provides an alternative method.
Procedure 5.10: Finding the Matrix of Inconveniently Defined Linear Transformation
Suppose T : R n → R m is a linear transformation. Suppose there exist vectors {⃗a 1 ,···,⃗a n } in R n
such that [ ⃗a 1 ··· ⃗a n
] −1 exists, and
T (⃗a i )= ⃗ b i
Then the matrix of T must be of the form
[ ⃗b1 ··· ⃗ bn
][
⃗a1 ··· ⃗a n
] −1
We will illustrate this procedure in the following example. You may also find it useful to work through
Example 5.9 using this procedure.
Example 5.11: Matrix of a Linear Transformation
Given Inconveniently
Suppose T : R 3 → R 3 is a linear transformation and
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡
1 0 0 2
T ⎣ 3 ⎦ = ⎣ 1 ⎦,T ⎣ 1 ⎦ = ⎣ 1
1 1 1 3
Find the matrix of this linear transformation.
Solution. By Procedure 5.10, A = ⎣
⎡
1 0 1
3 1 1
1 1 0
⎤
⎦
−1
⎤
and B = ⎣
⎡
⎦,T ⎣
⎡
1
1
0
⎤
0 2 0
1 1 0
1 3 1
Then, Procedure 5.10 claims that the matrix of T is
⎡
2 −2
⎤
4
C = BA −1 = ⎣ 0 0 1 ⎦
4 −3 6
Indeed you can first verify that T (⃗x)=C⃗x for the 3 vectors above:
⎡
⎦ = ⎣
⎤
⎦
0
0
1
⎤
⎦