Exercise 18, § 11

Let z = $\sum_H \alpha_H e_H$ be a non-zero pure p-vector of an n-dimensional vector space E, expressed using its components with respect to an arbitrary basis $(e_i)_{1 \leq i \leq n}$ of E. Let G be a subset of $[1, n]$ with p elements such that $a_G \neq 0$; let $(i_h)_{1 \leq h < p}$ be the sequence of indices in G arranged in increasing order and $(j_k)_{1 \leq k \leq n-p}$ the sequence of indices in the complement G' of G, arranged in increasing order. For every ordered pair $(h, k)$ of indices such that $1 \leq h \leq p$, $1 \leq k \leq n - p$, let $\beta_{hk}$ be the component $\alpha_H$ of z corresponding to the subset H of $[1, n]$ consisting of the $p - 1$ indices in G distinct from $i_h$ and of index $j_k \in G'$; let X be the matrix $(\beta_{hk})$ with p rows and $n - p$ columns. Given an arbitrary subset L of $[1, n]$ with p elements such that $L \cap G$ has $q \geq 1$ elements, show that $(a_G)^{q-1} a_L$ is equal, up to sign, to the minor of X of order q, consisting of the rows of index h such that $i_h \in G \cap L$ and the columns of index k such that $j_k \in L \cap G'$. (Write z as being of the form $a_G y_1 \wedge y_2 \wedge \cdots \wedge y_p$, where the vectors $y_i$ are such that, in the matrix Y with n rows and p columns whose columns are the $y_i$, the submatrix consisting of the rows whose indices belong to G is the unit matrix.)