A First Course in Linear Algebra, 2017a

9.6. Linear Transformations 493
2. Let⃗v 1 ,⃗v 2 ∈ U ∩W.Theninparticular,⃗v 1 ,⃗v 2 ∈ U. SinceU is a subspace, it follows that⃗v 1 +⃗v 2 ∈ U.
The same argument holds for W. Therefore ⃗v 1 +⃗v 2 is in both U and W and by definition is also in
U ∩W.
3. Let a be a scalar and ⃗v ∈ U ∩W. Then in particular, ⃗v ∈ U. SinceU is a subspace, it follows that
a⃗v ∈ U. The same argument holds for W so a⃗v is in both U and W. By definition, it is in U ∩W.
Therefore U ∩W is a subspace of V .
♠
ItcanalsobeshownthatU +W is a subspace of V.
We conclude this section with an important theorem on dimension.
Theorem 9.54: Dimension of Sum
Let V be a vector space with subspaces U and W . Suppose U and W each have finite dimension.
Then U +W also has finite dimension which is given by
dim(U +W)=dim(U)+dim(W) − dim(U ∩W)
{ }
Notice that when U ∩W = ⃗ 0 , the sum becomes the direct sum and the above equation becomes
dim(U ⊕W)=dim(U)+dim(W)
9.6 Linear Transformations
Outcomes
A. Understand the definition of a linear transformation in the context of vector spaces.
Recall that a function is simply a transformation of a vector to result in a new vector. Consider the
following definition.
Definition 9.55: Linear Transformation
Let V and W be vector spaces. Suppose T : V ↦→ W is a function, where for each ⃗x ∈ V,T (⃗x) ∈ W.
Then T is a linear transformation if whenever k, p are scalars and⃗v 1 and⃗v 2 are vectors in V
T (k⃗v 1 + p⃗v 2 )=kT (⃗v 1 )+pT (⃗v 2 )
Several important examples of linear transformations include the zero transformation, the identity
transformation, and the scalar transformation.