Prove or disprove the following; Let $V$ be a vector space and $U$ and $W$ two subspaces of $V$. If the set of vectors $\{b_1,\ldots b_n\}$ is a basis for $U\cap W$, then it is also a basis for both $U$ and $W$ themselves.
I am not even sure if this statement is true or not. I have tried to find a counter-example to disprove, which is what I feel needs to be done - but not sure.