A First Course in Linear Algebra, 2017a

492 Vector Spaces
9.5 Sums and Intersections
Outcomes
A. Show that the sum of two subspaces is a subspace.
B. Show that the intersection of two subspaces is a subspace.
We begin this section with a definition.
Definition 9.51: Sum and Intersection
Let V be a vector space, and let U and W be subspaces of V .Then
1. U +W = {⃗u +⃗w | ⃗u ∈ U and ⃗w ∈ W} andiscalledthesum of U and W .
2. U ∩W = {⃗v |⃗v ∈ U and⃗v ∈ W} andiscalledtheintersection of U and W .
Therefore the intersection of two subspaces is{ all } the vectors shared by both. If there are no vectors
shared by both subspaces, meaning that U ∩W = ⃗ 0 ,thesumU +W takes on a special name.
Definition 9.52: Direct Sum
{ }
Let V be a vector space and suppose U and W are subspaces of V such that U ∩W = ⃗ 0 . Then the
sum of U and W is called the direct sum and is denoted U ⊕W .
An interesting result is that both the sum U +W and the intersection U ∩W are subspaces of V.
Example 9.53: Intersection is a Subspace
Let V be a vector space and suppose U and W are subspaces. Then the intersection U ∩ W is a
subspace of V .
Solution. By the subspace test, we must show three things:
1. ⃗0 ∈ U ∩W
2. For vectors⃗v 1 ,⃗v 2 ∈ U ∩W,⃗v 1 +⃗v 2 ∈ U ∩W
3. For scalar a and vector⃗v ∈ U ∩W,a⃗v ∈ U ∩W
We proceed to show each of these three conditions hold.
1. Since U and W are subspaces of V , they each contain⃗0. By definition of the intersection,⃗0 ∈ U ∩W.