In mathematics, an affine combination of x1, ..., xn is a linear combination such that Here, x1, ..., xn can be elements (vectors) of a vector space over a field K , and the coefficients \alpha_{i} are elements of K . The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K . In this case the \alpha_{i} are elements of K (or \mathbb R for a Euclidean space), and the affine combination is also a point. See for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest affine space containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinations commute with any affine transformation T in the sense that In particular, any affine combination of the fixed points of a given affine transformation T is also a fixed point of T, so the set of fixed points of T forms an affine space (in 3D: a line or a plane, and the trivial cases, a point or the whole space). When a stochastic matrix, A, acts on a column vector, b, the result is a column vector whose entries are affine combinations of b with coefficients from the rows in A.

Related combinations

Affine geometry