= Basis of a Euclidean lattice
{title2=$\Lambda=B\mathbb Z^n$}
A <basis of a Euclidean lattice> $\Lambda\subset\mathbb R^n$ is a list of <linearly independent> <vectors> $b_1,\ldots,b_n$ such that
$$
\Lambda=\left\{\sum_{i=1}^n m_i b_i:m_i\in\mathbb Z\right\}.
$$
The 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> $\pm1$. The <covolume> is $|\det B|$; basis changes preserve it. This separates changing a description from changing the discrete set itself.
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