Subspace Intersection Problem: Proving W1 and W2 in R^n

[mlarson9000]

Homework Statement

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

Homework Equations

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.

 

[VeeEight]

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

 

[mlarson9000]

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

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.

 

[Mark44]

Wouldn't it have to be?

Well, if not, you're going to have a tough time proving it.

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?

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."

If so, I still don't know exactly how to say that in Math speak.

 

[HallsofIvy]

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.

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 ...