In mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is the semidirect product of Kⁿ and GL(n, k). It is a Lie group if K is the real or complex field.

A possible matrix representation of an affine transformation given by a pair

(M, v),

where M is an n×n matrix over K, and v a n×1 column vector, is the

(n + 1)×(n + 1)

matrix

(M*|v*).

Here M* is the (n + 1)×n matrix formed by adding a row of zeroes below M, and v* is the column matrix of size n + 1 formed by adding an entry 1 below v.

For an example of an affine group acting on a small finite geometry, see Geometry of the 4x4 Square.

Affine geometry | Group theory | Lie groups