Proving the Intersection of Subspaces is a Subspace: A Vector Space Proof

[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.

 

[TD]

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.

 

[TD]

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.