The near-ring of affine transformations

where a is variable in V, A is fixed in V, and T is a linear transformation of V. The transformation is nonsingular if and only if T is nonsingular. A few familiar facts 1 are: 1. The nonsingular transformations form the affine under iteration. 2. The translations, i.e. those A with T= I, form a normal subgroup of the group. 3. The quotient group of the group modulo the translation group is isomorphic to the group of nonsingular linear transformations. 4. If V is finite-dimensional with basis (a, , * , a.), let