Exercise 5, § 7

Let K be a commutative field, E, F two vector spaces over K and $u$ a linear mapping of E into F of rank r. Show that if $p \leq r$ the rank of $\bigwedge^p u$ is equal to $\binom{r}{p}$ and that if $p > r$, $\bigwedge^p u$ is identically zero (take in E a basis r vectors of which form a basis of a subspace supplementary to $u^{-1}(0)$ and the rest a basis of $u^{-1}(0)$).