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| In [[mathematics]], an '''affine combination''' of vectors ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> is a vector
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| :<math> \sum_{i=1}^{n}{\alpha_{i} \cdot x_{i}} = \alpha_{1} x_{1} + \alpha_{2} x_{2} + \cdots +\alpha_{n} x_{n}, </math>
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| called a [[linear combination]] of ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, in which the sum of the coefficients is 1, thus:
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| :<math>\sum_{i=1}^{n} {\alpha_{i}}=1. </math>
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| Here the vectors are elements of a given [[vector space]] ''V'' over a [[field (mathematics)|field]] ''K'', and the coefficients <math>\alpha _{i}</math> are [[scalar (mathematics)|scalars]] in ''K''.
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| This concept is important, for example, in [[Euclidean geometry]].
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| The act of taking an affine combination commutes with any [[affine transformation]] ''T'' in the sense that
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| :<math> T\sum_{i=1}^{n}{\alpha_{i} \cdot x_{i}} = \sum_{i=1}^{n}{\alpha_{i} \cdot Tx_{i}} </math> | |
| In particular, any affine combination of the [[fixed point (mathematics)|fixed point]]s of a given [[affine transformation]] <math>T</math> is also a fixed point of <math>T</math>, so the set of fixed points of <math>T</math> forms an [[affine subspace]] (in 3D: a line or a plane, and the trivial cases, a point or the whole space).
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| 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.
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| ==See also==
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| ===Related combinations===
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| {{details|Linear combination#Affine, conical, and convex combinations}}
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| *[[Convex combination]]
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| *[[Conical combination]]
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| *[[Linear combination]]
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| ===Affine geometry===
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| * [[Affine space]]
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| * [[Affine geometry]]
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| * [[Affine hull]]
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| ==References==
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| * {{Citation | last1=Gallier | first1=Jean | title=Geometric Methods and Applications | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-95044-0 | year=2001}}. ''See chapter 2''.
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| [[Category:Affine geometry]]
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