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
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
is also a fixed point of
, so the set of fixed points of
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.