# Affine combination

In mathematics, an **affine combination** of *x*_{1}, ..., *x*_{n} is a linear combination

such that

Here, *x*_{1}, ..., *x*_{n} can be elements (vectors) of a vector space over a field *K*, and the coefficients are elements of *K*.

The elements *x*_{1}, ..., *x*_{n} can also be points of a Euclidean space, and, more generally, of an affine space over a field *K*. In this case the are elements of *K* (or for a Euclidean space), and the affine combination is also a point. See Affine space § Affine combinations and barycenter for the definition in this case.

This concept is fundamental in Euclidean geometry and affine geometry, as the set of all affine combinations of a set of points form the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span.

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.

## See also

### Related combinations

### Affine geometry

## References

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