Let $\displaystyle U, V$ be finite dimensional linear spaces and let $\displaystyle \{u_1,...,u_n\}$ be a basis for $\displaystyle U$ and let $\displaystyle \{v_1,...,v_m\}$ be a basis for $\displaystyle V$. Let $\displaystyle W=U\times V$. Construct a basis for $\displaystyle W$.

I'm probably making this way harder than it really is. The only idea I came up with was

$\displaystyle \{(u_i, v_j) | 1\leq i\leq n, 1\leq j\leq m\}$

But I know that's wrong....