Category:Examples of Integer Lattices

This category contains examples of Integer Lattice.

Let $n$ be a positive integer.

Let $\R^n$ be the $n$-dimensional real Euclidean space.

Let $\CC$ be the Cartesian coordinate system of $n$ dimensions embedded in $\R^n$.

The integer lattice in $\R^n$ is the set of points in $\R^n$ which can be defined using coordinates of $\CC$ which are integers.

Thus it is the point lattice on $\R^n$ whose basis is the standard ordered basis over $\R^n$:

$\BB = \set {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$

where:

$\mathbf e_k := \tuple {\delta_{1 k}, \delta_{2 k}, \ldots, \delta_{n k} }$

$\delta_{j k}$ is the Kronecker delta.

Hence it is the point lattice whose basis is the set of $n$ vectors:

$\BB = \set {\tuple {1, 0, 0, \ldots, 0}, \tuple {0, 1, 0, \ldots, 0}, \tuple {0, 0, 1, \ldots, 0}, \ldots, \tuple {0, 0, 0, \ldots, 1} }$