Affine combination

In mathematics, an affine combination of vectors x1, …, xn is a vector

called a linear combination of x1, …, xn, in which the sum of the coefficients is 1, thus:

Here the vectors are elements of a given vector space V over a field K, and the coefficients






α




i






{\displaystyle \alpha _{i}}


are scalars in K.

This concept is important, for example, in Euclidean geometry.

The act of taking an affine combination commutes with any affine transformation T in the sense that

In particular, any affine combination of the fixed points of a given affine transformation





T




{\displaystyle T}


is also a fixed point of





T




{\displaystyle T}


, so the set of fixed points of





T




{\displaystyle T}


forms an affine subspace (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.