In mathematics, a linear combination of vectors x1, ..., xn
is called an affine combination of x1, ..., xn when the sum of the coefficients is 1, that is,
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 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 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.