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Concept
Affine combination
Summary
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 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 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, , the result is a column vector whose entries are affine combinations of with coefficients from the rows in A.
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