# Subspace problem

1. Jul 9, 2009

### mlarson9000

1. The problem statement, all variables and given/known data

Let W1 and W2 be two subspaces of R^n. Prove that their intersection is also a subspace.

2. Relevant equations

3. The attempt at a solution

I know that in R^2 and R^3 the intersection would be the origin, which would be the zero vector, which would be a subspace, but I don't know how to make a general argument about this.

2. Jul 9, 2009

### VeeEight

Let W be the intersection of W1 and W2. Is the vector space W closed under addition and scalar multiplication?

3. Jul 10, 2009

### mlarson9000

Wouldn't it have to be? If W is in both W1 and W2, which are both subspaces and therefore closed under addition and scalar multiplication, wouldn't W be also? If so, I still don't know exactly how to say that in Math speak.

4. Jul 10, 2009

### Staff: Mentor

Well, if not, you're going to have a tough time proving it.
Take a couple of arbitrary vectors u1 and u2 in W, and show that their sum is also in W. Then take an arbitrary scalar s, and show that su1 is in W. That's how you would do it it "math speak."

5. Jul 10, 2009

### HallsofIvy

Staff Emeritus
Suppose u and v are in W. Then u and v are both is W1 and, since W1 is a subspace, u+ v is in W1. Also u and v are both in W2 ....