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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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