Affine map

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 a geometric setting, 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, .., v_n are linearly dependent if scalars a₁, a₂, .., a_n exist such that a₁v₁ + ... + a_nv_n = 0 and not all of these scalars are 0. Similarly they are affinely dependent if the same is true and also a₁ + ... + a_n = 0. Such a vector (a₁, ..., a_n) is an affine dependence among the vectors v₁, v₂, ..., v_n.

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

<math>\begin{bmatrix}

1&0&0&0&1&1&1&1 \\ 1&1&0&0&0&1&1&1 \\ 1&1&1&0&0&0&1&1 \\ 1&1&1&1&0&0&0&1 \\ 1&1&1&1&1&0&0&0 \\ 0&1&1&1&1&1&0&0 \\ 0&0&1&1&1&1&1&0 \\ 0&0&0&1&1&1&1&1\end{bmatrix}<math>

and {v} is the vector

<math>\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}<math>

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:

<math>a_0' = a_0 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 1<math>

<math>a_1' = a_0 \oplus a_1 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 0<math>

<math>a_2' = a_0 \oplus a_1 \oplus a_2 \oplus a_6 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1<math>

<math>a_3' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1<math>

<math>a_4' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 = 0<math>

<math>a_5' = a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus 1 = 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 = 1<math>

<math>a_6' = a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 = 1<math>

<math>a_7' = a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 0 = 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1<math>

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

See also

affine geometry, homothety, similarity transformation.