Category:Definitions/Point Lattices

This category contains definitions related to Point Lattices.
Related results can be found in Category:Point Lattices.

Definition 1

A point lattice is a discrete subgroup of $\R^m$ under addition.

Definition 2

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

Let $\set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n}$ be a linearly independent set of vectors of $\R^m$.

A point lattice in $\R^m$ is the set of all integer linear combinations of such vectors.

That is:

$\ds \map \LL {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n} = \set {\sum_{i \mathop = 1}^n a_i \mathbf v_i : a_i \in \Z}$