Exercise 19, § 11

Over a finite-dimensional vector space E, let $z = x_1 \wedge x_2 \wedge \cdots \wedge x_p$ be a pure p-vector $\neq 0$; let $v^*$ be a q-form ($q \leq p$) and $u^*$ an arbitrary element of $\bigwedge(E^*)$. Show that

where H runs through the set of subsets of $[1, p]$ with q elements, K is the complement of H in $[1, p]$ and $x_H$ (resp. $x_K$) denotes the exterior product of the $x_i$ such that $i \in H$ (resp. $i \in K$).