http://www.mathhelpforum.com/math-he...oup-proof.html, try to understand it. The proof to your problem is almost exactly the same.
An alternate way to think of it is like this. are subspaces, thus, they are abelian groups under vector addition (definition of vector spaces). Thus, by the conditions of the problem is a subspace and hence an abelian group. The link I gave you says, the union of two subgroups is a subgroup if and only if one is contained in the other. Now since is an abelian group under addition of vectors follows that one of them is contained in the other. Thus, or