TITLE: Geometric meaning of unimodular matrix
QUESTION [10 upvotes]: Rotations are given by unitary matrices.
What is the geometric meaning of unimodular matrices that are not unitary?

REPLY [8 votes]: If the matrix has integer elements, then the geometric meaning is that a unimodular transformation maps the integer lattice onto itself:

Consider a basis $B$ of an $m$-dimensional lattice
  $L(B)=\{Bx:x\in\mathbb{Z}^m$}, and another basis $C$, then $L(B)=L(C)$
  if and only if there exists a unimodular matrix $M$ (an $m\times m$
  matrix with integer entries and determinant $\pm 1$) such that $B=CM$.