Let $V = V_1 \oplus V_2$. Also, $V$ equals the direct sum of $V_3$ and $V_4$.
$V_1$, $V_2$, $V_3$, $V_4$ are subspaces of $V$.
Prove or disprove that:
$$ V = (V_1 \cap V_3) \oplus (V_1 \cap V_4) \oplus (V_2 \cap V_3) \oplus (V_2 \cap V_4) $$
Not sure how to start on this one. I know that the intersection of $V_1$ and $V_2$ is the zero vector, likewise for $V_3$ and $V_4$. Not sure how this helps.