Let $V$ be a real vector space and $W_1$ and $W_2$ two finite dimensional subspaces. Let $(u_1,\ldots, u_m)$ be a basis for $W_1$ and $(v_1,\ldots, v_n)$ be a basis for $W2$.
Suppose that $(u_1,\ldots, u_m, v_1,\ldots, v_n)$ is linearly independent. Prove that $W_1 ∩ W_2 = \{0\}$.
I know that if $w ∈ W_1 ∩ W_2$, then $w$ is a linear combination of the $u_i$ and also a linear combination of the $v_j$). I am however not sure how to proceed next.