A First Course in Linear Algebra, 2017a

494 Vector Spaces
Example 9.56: Linear Transformations
Let V and W be vector spaces.
1. The zero transformation
0:V → W is defined by 0(⃗v)=⃗0 for all⃗v ∈ V.
2. The identity transformation
1 V : V → V is defined by 1 V (⃗v)=⃗v for all⃗v ∈ V .
3. The scalar transformation Let a ∈ R.
s a : V → V is defined by s a (⃗v)=a⃗v for all⃗v ∈ V .
Solution. We will show that the scalar transformation s a is linear, the rest are left as an exercise.
By Definition 9.55 we must show that for all scalars k, p and vectors ⃗v 1 and ⃗v 2 in V, s a (k⃗v 1 + p⃗v 2 )=
ks a (⃗v 1 )+ps a (⃗v 2 ). Assume that a is also a scalar.
s a (k⃗v 1 + p⃗v 2 ) = a(k⃗v 1 + p⃗v 2 )
= ak⃗v 1 + ap⃗v 2
= k (a⃗v 1 )+p(a⃗v 2 )
= ks a (⃗v 1 )+ps a (⃗v 2 )
Therefore s a is a linear transformation.
♠
Consider the following important theorem.
Theorem 9.57: Properties of Linear Transformations
Let V and W be vector spaces, and T : V ↦→ W a linear transformation. Then
1. T preserves the zero vector.
T (⃗0)=⃗0
2. T preserves additive inverses. For all⃗v ∈ V ,
T (−⃗v)=−T (⃗v)
3. T preserves linear combinations. For all⃗v 1 ,⃗v 2 ,...,⃗v m ∈ V and all k 1 ,k 2 ,...,k m ∈ R,
T (k 1 ⃗v 1 + k 2 ⃗v 2 + ···+ k m ⃗v m )=k 1 T (⃗v 1 )+k 2 T (⃗v 2 )+···+ k m T (⃗v m ).
Proof.
1. Let⃗0 V denote the zero vector of V and let⃗0 W denote the zero vector of W. We want to prove that
T (⃗0 V )=⃗0 W .Let⃗v ∈ V .Then0⃗v =⃗0 V and
T (⃗0 V )=T (0⃗v)=0T (⃗v)=⃗0 W .