= Unimodular change of lattice basis
{title2=$B_2=B_1U,\quad U\in\mathrm{GL}_n(\mathbb Z)$}
Let $B_1,B_2$ be real square <invertible matrices> whose columns are <lattice bases>. They generate the same <Euclidean lattice> precisely when
$$
\boxed{B_1\mathbb Z^n=B_2\mathbb Z^n
\quad\Longleftrightarrow\quad
B_2=B_1U\text{ for a unimodular matrix }U.}
$$
<Intro to Lattice Algs & Crypto, Mastermath Spring 2018/Lecture 1: Introduction>[§7, Lemma 13, printed p. 9: equivalent lattice bases]{ref}
If $B_2=B_1U$, multiplication by $U$ merely permutes the integer coordinate vectors, so the two generated <sets> agree. Conversely, equality of the generated <sets> means every column of either basis is an <integer linear combination> of the other. Hence $B_2=B_1U$ and $B_1=B_2V$ for <integer matrices> $U,V$. Substitution followed by multiplication by $B_1^{-1}$ gives $UV=I$, so the <integer inverse criterion for a unimodular matrix> applies.
Equal real <linear spans> are a weaker condition: any two real square <invertible matrices> span the same ambient <vector space>. Even equal <covolumes> do not force the same <Euclidean lattice>. The requirement that the change matrix and its <inverse matrix> have integer entries provides the missing arithmetic information.
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