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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. Intuitively, these are precisely the functions that map straight lines to straight lines.

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).

## Example of an affine transformation

The following equation expresses an affine transformation in GF(2) (with "+" representing XOR):

{a'} = [M]{a} + {v}

where [M] is the matrix

and {v} is the vector

For instance, the affine transformation of the element {a} = x7 + x6 + x3 + x = {11001010} in big-endian binary notation = {CA} in big-endian hexadecimal notation, is calculated as follows:

Thus, {a'} = x7 + x6 + x5 + x3 + x2 + 1 = {11101101} = {ED}