- #1

bornofflame

- 56

- 3

## 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?