\begin{definition}[Definition:Covolume of Lattice]
Let $L$ be a lattice in $\R^n$.
Let $\left({v_1, \ldots, v_n}\right)$ be an ordered basis for $L$.
Let $v_i = \left({v_{i 1}, \ldots , v_{i n} }\right)$ for $i \in \left\{ {1, \ldots, n}\right\}$.
The '''covolume''' of $L$ is the determinant of the matrix:
:$\begin{bmatrix}
v_{1 1} & v_{1 2} & \cdots & v_{1 n} \\
v_{2 1} & v_{2 2} & \cdots & v_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
v_{n 1} & v_{n 2} & \cdots & v_{n n} \\
\end{bmatrix}$
\end{definition}