Proposition 15

Let $\mathbf{E}$ be a vector space over a commutative field; for $p$ vectors $x_i \in \mathbf{E}$ ($1 \leq i \leq p$) to be linearly independent, it is necessary and sufficient that $x_1 \wedge x_2 \wedge \cdots \wedge x_p \neq 0$.