Determinant as signed volume

Summary

Let $n \ge 1$ and let $v_1, \dots, v_n \in \R^n$ be the columns of a real $n \times n$ matrix $A$. The parallelepiped spanned by them is $P = \{t_1 v_1 + \cdots + t_n v_n : 0 \le t_i \le 1\}$. Its $n$-dimensional volume is $\abs{\det A}$, where $\det$ is the determinant. If $\det A \ne 0$, then its sign is positive when $(v_1, \dots, v_n)$ has the orientation of the standard basis, and negative when it has the opposite orientation.