We are given a vector space V of dimension 5
We are given that it has two subspaces U & W, both of dimension 3.
We are to prove that U intersects W at a vector other than 'zero'
My progress
Well, I believe since both of them have dimension 3, their basis must intersect at some point. I just don't understand how to prove it.
Best Answer
Assume that $U,W \subseteq V$ intersect as $0$ only. Then the direct sum $U \oplus W$ have dimension $6$. (Try to prove this) But $U \oplus W$ is a vector subspace of $V$, which means its dimension cannot exceed $\dim V$. Threfore $U$ and $W$ must intersect somewhere else.