# Affine combination

In mathematics, a **linear combination** of vectors *x*_{1}, ..., *x*_{n}

is called an **affine combination** of *x*_{1}, ..., *x*_{n} 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.

## 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*.