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:
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.
- Gallier, Jean (2001), Geometric Methods and Applications, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95044-0. See chapter 2.