Prove or give a counterexample: If $v_1,v_2,v_3,v_4$ is a basis of $V$ and $U$ is a subspace of $V$ such that $v_1,v_2 \in U$ and $v_3,v_4 \notin U$, then $v_1,v_2$ is a basis of $U$.

Counterexample: Let $V = \mathbf R^4$ and

Let our basis $v_1,v_2,v_3,v_4$ of $V$ be

(This is the same as 2b/5 so I won't waste time proving it's a basis, later on we'd say they're isomorphic)

We have $v_1,v_2 \in U$ and $v_3,v_4 \notin U$ as desired, but $v_1,v_2$ don't span $U$ as they don't control the third component. Thus $v_1,v_2$ is not a basis, completing our counterexample.