 Basis of a Euclidean lattice

A basis of a Euclidean lattice $\Lambda \subset {\mathbb{R}}^n$ is a list of linearly independent vectors $b_1,\dots ,b_n$ such thatThe coefficients must be integers: a real basis spanning the same ambient vector space need not generate the same Euclidean lattice.

For ${\mathbb{Z}}^2$, the vectors $(1,0),(1,1)$ form a lattice basis, because $(a,b)=(a-b)(1,0)+b(1,1)$. The independent pair $(2,0),(0,1)$ instead generates only points with even first coordinate. Writing basis vectors as columns of $B$, another matrix $BU$ is a basis of the same lattice exactly when $U$ is an integer unimodular matrix, with determinant $\pm 1$. The covolume is $|\det B|$; basis changes preserve it. This separates changing a description from changing the discrete set itself.