Unimodular transformation

A linear transformation of a finite-dimensional vector space whose matrix has determinant $\pm 1$.

Comments

The name "unimodular transformation" is often restricted to mean a linear transformation with determinant $1$. In the context of a vector space $V$ over a field $k$ which is the quotient field of an integral domain $D$, with a fixed $k$- basis $a _ {1} \dots a _ {n}$ in $V$, a linear transformation is called unimodular if its matrix with respect to $a _ {1} \dots a _ {n}$ has entries in $D$ and determinant a unit in $D$. Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant $1$ one often calls this the unimodular group, or, more commonly nowadays, the special linear group.