affine combination

{{Short description|Linear combination whose coefficients sum to 1}}

In mathematics, an affine combination of {{math|x₁, ..., x_n}} is a linear combination

:$\backslash sum\_\{i=1\}^\{n\}\{\backslash alpha\_\{i\}\; \backslash cdot\; x\_\{i\}\}\; =\; \backslash alpha\_\{1\}\; x\_\{1\}\; +\; \backslash alpha\_\{2\}\; x\_\{2\}\; +\; \backslash cdots\; +\backslash alpha\_\{n\}\; x\_\{n\},$

such that

:$\backslash sum\_\{i=1\}^\{n\}\; \{\backslash alpha\_\{i\}\}=1.$

Here, {{math|x₁, ..., x_n}} can be elements (vectors) of a vector space over a field {{math|K}}, and the coefficients $\backslash alpha\_\{i\}$ are elements of {{math|K}}.

The elements {{math|x₁, ..., x_n}} can also be points of a Euclidean space, and, more generally, of an affine space over a field {{math|K}}. In this case the $\backslash alpha\_\{i\}$ are elements of {{math|K}} (or $\backslash mathbb\; R$ for a Euclidean space), and the affine combination is also a point. See {{slink|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 {{math|T}} in the sense that

:$T\backslash sum\_\{i=1\}^\{n\}\{\backslash alpha\_\{i\}\; \backslash cdot\; x\_\{i\}\}\; =\; \backslash sum\_\{i=1\}^\{n\}\{\backslash alpha\_\{i\}\; \backslash cdot\; Tx\_\{i\}\}.$

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 space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).

When a stochastic matrix, {{mvar|A}}, acts on a column vector, {{vec|b}}, the result is a column vector whose entries are affine combinations of {{vec|b}} with coefficients from the rows in {{mvar|A}}.