Affine combination

Linear combination whose coefficients sum to 1

In mathematics, an affine combination of x1, ..., xn is a linear combination

∑ i = 1 n α i ⋅ x i = α 1 x 1 + α 2 x 2 + ⋯ + α n x n , {\displaystyle \sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n},}

such that

∑ i = 1 n α i = 1. {\displaystyle \sum _{i=1}^{n}{\alpha _{i}}=1.}

Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients  α i {\displaystyle \alpha _{i}} are elements of K.

The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the  α i {\displaystyle \alpha _{i}} are elements of K (or  R {\displaystyle \mathbb {R} } 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, because the set of all affine combinations of a set of points forms the smallest affine space 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

T ∑ i = 1 n α i ⋅ x i = ∑ i = 1 n α i ⋅ T x i . {\displaystyle T\sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\sum _{i=1}^{n}{\alpha _{i}\cdot Tx_{i}}.}

In particular, any affine combination of the fixed points of a given affine transformation T {\displaystyle T} is also a fixed point of  T {\displaystyle T} , so the set of fixed points of  T {\displaystyle T} forms an affine space (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

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### Related combinations

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Further information: Linear combination § Affine, conical, and convex combinations

* Convex combination
* Conical combination
* Linear combination

### Affine geometry

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* Affine space
* Affine geometry
* Affine hull

## References

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

## External links

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* Notes on affine combinations.