A First Course in Linear Algebra, 2017a

278 Linear Transformations
The necessary augmented matrix and resulting reduced row-echelon form are given by:
⎡ ⎤ ⎡ ⎤
1 4 −7
1 0 1
⎣ 3 0 3 ⎦ →···→⎣
0 1 −2 ⎦
1 5 −9
0 0 0
Hence a = 1,b = −2 and
⎡
⎣
−7
3
−9
⎤
⎡
⎦ = 1⎣
1
3
1
⎤
⎡
⎦ +(−2) ⎣
4
0
5
⎤
⎦
Now, using the third property above, we have
⎡ ⎤ ⎛ ⎡ ⎤ ⎡
−7
1
T ⎣ 3 ⎦ = T ⎝1⎣
3 ⎦ +(−2) ⎣
−9
1
⎡ ⎤ ⎡ ⎤
1 4
= 1T ⎣ 3 ⎦ − 2T ⎣ 0 ⎦
1 5
⎡
Therefore, T ⎣
−7
3
−9
⎤
⎦ =
⎡
⎢
⎣
−4
−6
2
−12
⎤
⎥
⎦ .
=
=
⎡
⎢
⎣
⎡
⎢
⎣
4
4
0
−2
−4
−6
2
−12
⎤ ⎡
⎥
⎦ − 2 ⎢
⎣
⎤
⎥
⎦
4
5
−1
5
⎤
⎥
⎦
4
0
5
⎤⎞
⎦⎠
♠
Suppose two linear transformations act in the same way on ⃗x for all vectors. Then we say that these
transformations are equal.
Definition 5.15: Equal Transformations
Let S and T be linear transformations from R n to R m .ThenS = T if and only if for every⃗x ∈ R n ,
S(⃗x)=T (⃗x)
Suppose two linear transformations act on the same vector ⃗x, first the transformation T andthena
second transformation given by S. Wecanfindthecomposite transformation that results from applying
both transformations.