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.