Let U and V be subspaces of a vector space W. Prove that their intersection U ∩ V is also a subspace of W.
Get to work on that.
It is trivial to observe that the elements in the intersection are contained in the Vector Space.
You have three things to show:
1) There is a zero vector.
2) Closed for vector addition.
3) Closed for scalar multiplication.
Let's see how you do it.