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

\( \sum _{{i=1}}^{{n}}{\alpha _{{i}}\cdot x_{{i}}}=\alpha _{{1}}x_{{1}}+\alpha _{{2}}x_{{2}}+\cdots +\alpha _{{n}}x_{{n}}, \)

such that

\( \sum _{{i=1}}^{{n}}{\alpha _{{i}}}=1. \)

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

\( {\displaystyle T\sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\sum _{i=1}^{n}{\alpha _{i}\cdot Tx_{i}}.} \)

In particular, any affine combination of the fixed points of a given affine transformation T is also a fixed point of T, so the set of fixed points of T 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

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

Convex combination

Conical combination

Linear combination

Affine geometry

Affine space

Affine geometry

Affine hull

References

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

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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