Fundamentals of Matrix Algebra, 2011a

Chapter 5
Graphical Exploraons of Vectors
.
Ḋefinion 29
Linear Transformaon
A transformaon T : R n → R m is a linear transformaon if
it sasfies the following two properes:
.
1. T(⃗x +⃗y) = T(⃗x) + T(⃗y) for all vectors ⃗x and⃗y, and
2. T(k⃗x) = kT(⃗x) for all vectors ⃗x and all scalars k.
If T is a linear transformaon, it is oen said that “T is linear.”
Let’s learn about this definion through some examples.
. Example 98 Determine whether or not the transformaon T : R 2 → R 3 is a
linear transformaon, where
⎡ ⎤
([ ])
x1
T = ⎣ x2 1
2x
x 1
⎦ .
2
x 1 x 2
S
We’ll arbitrarily pick two vectors ⃗x and⃗y:
[ ]
[ ]
3 1
⃗x = and ⃗y = .
−2 5
Let’s check to see if T is linear by using the definion.
1. Is T(⃗x +⃗y) = T(⃗x) + T(⃗y)? First, compute ⃗x +⃗y:
[ ] [ ] [ ]
3 1 4
⃗x +⃗y = + = .
−2 5 3
204
Now compute T(⃗x), T(⃗y), and T(⃗x +⃗y):
([ ]) 3
T(⃗x) = T
−2
⎡ ⎤
9
= ⎣ 6
−6
⎦
Is T(⃗x +⃗y) = T(⃗x) + T(⃗y)?
⎡
([ ]) 1
T(⃗y) = T
5
⎡ ⎤
1
= ⎣ 2 ⎦
5
⎣ 9 6
−6
⎤
⎦ +
⎡
⎣ 1 2
5
Therefore, T is not a linear transformaon. .
⎤
⎦ !
≠
⎡
⎣ 16
8
12
⎤
⎦ .
([ ]) 4
T(⃗x +⃗y) = T
3
⎡ ⎤
16
= ⎣ 8 ⎦
12