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 a Lie group if K is the real or complex field.

There is more than one convenient way to describe the structure of affine groups. There is the abstract result that it is a semidirect product: this is given on the affine space page. There is also a more down-to-earth matrix representation: represent a pair

(M, v),

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