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Homework Statement
From Linear Algebra and Its Applications, 5th Edition, David Lay
Chapter 4, Section 1, Question 32
Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.) Give an example in ℝ^{2} to show that the union of two subspaces is not, in general, a subspace.
Homework Equations
/theorems[/B]Theorem 1: If v_{1},...v_{p} are in vector space V, then Span{v_{1},...v_{p}} is a subspace of V.
The Attempt at a Solution
This is what I started off with:
Let u,v ∈ H; s, t ∈ K
0v ∈ H, K
u + v ∈ H
s + t ∈ K
In the middle of writing that down, I was thinking that v is a set of vectors in V, and that H ∩ K = Span{v}, therefore, per Theorem 1 in the book, H ∩ K is a subspace of V.
Written as:
v ∈ V
H ∩ K = Span{v}
∴ H ∩ K is a subspace of V per Theorem 1.
For the second part I used the following:
m, n ∈ ℝ^{2}
m = {[x, y]: x, y ≥ 0}
n = {[x, y]: x ≤ 0, y ≥ 0}
m ∪ n = {[x, y]: x = ℝ, y ≥ 0}
let c = 1, u = [1, 3]
cu = [1, 3]
cu ∉ m ∪ n
∴ m ∪ n is not a subspace
Is my train of thinking correct concerning this problem?
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