Is the Intersection of Two Subspaces Also a Subspace?

[forty]

Let H and K be subspaces of a vector space V. Prove that the intersection K and H is a subspace of V.

Intuitively I can see that this is true... Both H and K must be closed under vector addition and scalar multiplication so there intersection must also be closed under both those.

How do i prove this mathematically. And is what I've even said correct?

Thanks :-D

 

[Hurkyl]

And is what I've even said correct?

Try some specific examples to get some empirical validation of your conjecture, or to look for a counterexample.

(Gives you time to do this)

Assuming it checks out, we can answer your question by trying to prove it mathematically!

Intuitively I can see that this is true... Both H and K must be closed under vector addition and scalar multiplication so there intersection must also be closed under both those.

How do i prove this mathematically.

Definitions are almost always a very good place to start. And since you're learning linear algebra, it's probably a good idea to try and translate the problem into algebraic statements.

 

[forty]

I really have no idea where to begin... how would I write something like that in an algebraic form?

 

[Mark44]

If K was a subset of a vector space V, how would you go about showing that K was a subspace of V? I know that's not the question you're working on, but maybe it will get you thinking in the right way.