A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. An affine combination is a linear combination in which the sum of the coefficients is 1.
An affine subspace of a vector space is a coset of a linear subspace; i.e., it is the result of adding a constant vector to every element of the linear subspace. A linear subspace of a vector space is a subset that is closed under linear combinations; an affine subspace is one that is closed under affine combinations.
Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three. Vectors v1,v2,..,vn are linearly dependent if scalars a1,a2,..,an exist such that a1v1+...+anvn=0 and not all of these scalars are 0. Similarly they are affinely dependent if the same is true and also a1+...+an=0. Such a vector (a1,...,an) is an affine dependence among the vectors v1,v2,..,vn.
The set of all affine transformations forms a group under the operation of composition of functions. That group is called the affine group, and is the semidirect product of Kn and GL(n, k).
The following equation expresses an affine transformation in GF(2) (with "+" representing XOR):
Example of an affine transformation
where [M] is the matrix
See also: affine geometry, homothety, similarity transformation.