Affine transformation

An affine transformation or affine map 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,..,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.