Lattices in Rn

Definition: A Lattice is a set $\Lambda$ of the form $\Lambda = A \mathbb{Z}^n = \left \{ x_1 \vec{a_1} + x_2 \vec{a_2} + ... + x_n \vec{a_n} : (x_1, x_2,..., x_n) \in \mathbb{Z}^n \right \}$ where $A$ is an $n \times n$ invertible matrix. The points in $\Lambda$ are called Lattice Points.

If $A = I$ is the $2 \times 2$ identity matrix then the lattice points of $\Lambda = I\mathbb{Z}^2 = \mathbb{Z}^2$ are simply the points $(x, y) \in \mathbb{R}^2$ for which $x, y \in \mathbb{Z}$:

In general, if $A$ is an $n \times n$ invertible matrix then $A$ acts as a linear transformation on $\mathbb{Z}^n$, and so the lattice points of $\Gamma$ are obtained by linearly transforming each lattice point in $I\mathbb{Z}^n$ by $A$.

Definition: Let $\Lambda = A \mathbb{Z}^n$ be a lattice. The Determinant of $\Lambda$ is defined to be $d(\Lambda) = |\det(A)|$.