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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... This picture illustrates how the hours in a clock form a group. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...

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.