Why is determinant called volume of the fundamental parallelepiped in geometry of numbers?

Question

Let $v_1, ..., v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$. Then they form a lattice $\Lambda \subseteq \mathbb{R}^n$ and the volume of the fundamental domain is $|\det A|$, where $A$ is the matrix obtained by putting $v_i$'s into columns. Could someone explain me why we call this determinant a "volume"?

Answer 1

If you form the parallelepiped by $\{\sum_i c_i v_i \, | \, 0 \leq c_i \leq 1\}$ then the geometric volume of this object in $n$ dimensions is precisely $\det V$ where $V$ is the matrix with column vectors given by the $v_i$. If you'd like an intuitive understanding, note that it works for a cube and it also works for a "sheared" cube where you shift one dimension to the right or left in some axis-perpendicular direction. Also if $\det V = 0$ then the vectors are linearly dependent so the "parallepiped" lies in a lower dimensional hyperplane so the volume in dimension $n$ is $0$.