Exercise 14, § 7

Let M be a projective A-module and let $x_1, \ldots, x_n$ be linearly independent elements of M; for every subset $H = \{i_1, \ldots, i_m\}$ of $m \leq n$ elements of the interval $\{1, n\}$, where $i_1 < i_2 < \cdots < i_m$, we write

Show that when H runs through the set of subsets of $\{1, n\}$ with m elements, the $x_H$ are linearly independent in $\Lambda(M)$.