Proposition 15

Let E be a vector space of dimension n and let e be an element $\neq 0$ of $\wedge^n(E)$ (hence forming a basis of this vector space). Let $\phi : \wedge(E) \to \wedge(E^*)$ be the vector space isomorphism associated with e (no. 11, Proposition 12). If z is a pure element of $\wedge^p(E)$, then $\phi(z)$ is a pure element of $\wedge^{n-p}(E^*)$ and the subspaces associated with z and $\phi(z)$ are orthogonal.

The cases $p = 0$ and $p = n$ are trivial. Suppose therefore that $1 \leq p \leq n - 1$ and let $z = x_1 \wedge \cdots \wedge x_p \neq 0$. Then there exists a basis $(e_i)_{1 \leq i \leq n}$ of E such that $e_i = x_i$ for $1 \leq i \leq p$ and $e = e_1 \wedge e_2 \wedge \cdots \wedge e_n$. It then follows from formula (78) of no. 11 that $\phi(z) = \pm e_{p+1}^* \wedge \cdots \wedge e_n^*$, whence the proposition.