What is "affine combination"

n. (context linear algebra English) A linear combination (of vectors in Euclidean space) in which the coefficients all add up to one.

In mathematics, an affine combination of vectors x, ..., x is a vector

∑α ⋅ x = αx + αx + ⋯ + αx, 
called a linear combination of x, ..., x, in which the sum of the coefficients is 1, thus:

∑α = 1.

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

T∑α ⋅ x = ∑α ⋅ Tx
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 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.