Exercise 15, § 11

Let E be a free A-module of dimension $n$. Let $z$ be a $p$-vector over E and $z^*$ a $(p + q)$-form on E; $z$ can be canonically identified ($§ 7$, Exercise 8) with the skewsymmetrization $az_1$ of a contravariant tensor $z_1$ of order $p$ and $z^*$ with the skewsymmetrization of a covariant tensor of order $p + q$. If, in the mixed tensor $z_1 . z^*$, the $k$-th contravariant index and the $(p + k)$-th covariant index are contracted for $1 \leq k \leq p$, show that the covariant tensor of order $q$ thus obtained is skewsymmetrized and is canonically identified with the $q$-form $z^* \wedge z$, up to sign.