structural:scholar_edition, stylistic:tone
Groups

As an affine transformation is invertible, the square matrix A {\displaystyle A} appearing in its matrix representation is invertible. The matrix representation of the inverse transformation is thus

The invertible affine transformations (of an affine space onto itself) form the affine group, which has the general linear group of degree n {\displaystyle n} as subgroup and is itself a subgroup of the general linear group of degree n + 1 {\displaystyle n+1}.

The similarity transformations form the subgroup where A {\displaystyle A} is a scalar times an orthogonal matrix. For example, if the affine transformation acts on the plane and if the determinant of A {\displaystyle A} is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance.

Each of these groups has a subgroup of orientation -preserving or positive affine transformations: those where the determinant of A {\displaystyle A} is positive. In the last case this is in 3D the group of rigid transformations (proper rotations and pure translations).

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context.