Affine combination

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

such that

Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients 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 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

Affine geometry


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