Exercise 8, § 11

Let E be a free A-module of dimension $n$. For every A-linear mapping $v$ of $\Lambda(M)$ into an A-module G, we can write $v = \sum_{p=0}^n v_p$, where $v_p$ is the restriction of $v$ to $\Lambda^p(M)$; we then set $\eta v = \sum_{p=0}^n (-1)^p v_p$, so that $\eta^2 v = v$. Let $u$ be an isomorphism of M onto its dual $M^*$ and let $\Delta$ be the determinant of the matrix of $u$ with respect to a basis $(e_i)$ of M and the dual basis of $M^*$ (a determinant which does not depend on the basis of M chosen). If $\phi$ is the isomorphism of $\Lambda(M)$ onto $\Lambda(M^*)$ defined starting with $e = e_1 \wedge e_2 \wedge \cdots \wedge e_n$, show the formula