# Intersection of subspaces

Benny
Hi can someone please help me with the following question. Such questions always trouble me because I don't know where to start and/or cannot continue after starting.

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

By the way that the question is set out I figure that all I need to show is that the intersection of K and H is non-empty, closed under scalar multiplication and addition. So here is what I've tried.

H and K are subspaces of the vector space V so they both contain the zero vector. So it follows that the intersection contains the zeor vector so that $$H \cap K \ne \emptyset$$.

That's all I can think of. I'm not sure if I can make any other assumptions about vectors which are common to H and K and so I cannot continue.

Homework Helper
Now show that a lineair combination of vectors of $$H \cap K$$ is still in $$H \cap K$$.

Benny
Thanks for your response but that's the sort of thing that I'm having trouble with. All I've been able to show is that the zero vector is in the intersection. I don't know which are vectors are in the intersection. I cannot figure out what else I extract from the stem of the question to assist me. It's probably just a conceptual thing but I can't really see what I can and can't use.

Homework Helper
Well you already showed it's not empty. Now take the scalars $\alpha ,\beta \in \mathbb{R}$ (or any other field of course) and the vectors [itex]\vec x,\vec y \in H \cap K[/tex]. Now, since the vectors are in both subspaces, we can say that:
$$\alpha \vec x + \beta \vec y \in H$$
$$\alpha \vec x + \beta \vec y \in K$$

And thus: $$\alpha \vec x + \beta \vec y \in H \cap K$$

Benny
Ok thanks for the help.