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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> |
| | 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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| == Scottrade Cheap Longchamp Bags In Singapore == | | The act of taking an affine combination commutes with any [[affine transformation]] ''T'' in the sense that |
| | :<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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| == A new site to compare Shop Michael Kors Australia == | | ==See also== |
| | ===Related combinations=== |
| | {{details|Linear combination#Affine, conical, and convex combinations}} |
| | *[[Convex combination]] |
| | *[[Conical combination]] |
| | *[[Linear combination]] |
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| | ===Affine geometry=== |
|
| | * [[Affine space]] |
| <li>[http://audeladeleau.free.fr/spip.php?article11/ http://audeladeleau.free.fr/spip.php?article11/]</li>
| | * [[Affine geometry]] |
|
| | * [[Affine hull]] |
| <li>[http://xiaocaopingyu.com/discuz/forum.php?mod=viewthread&tid=480991&fromuid=38719 http://xiaocaopingyu.com/discuz/forum.php?mod=viewthread&tid=480991&fromuid=38719]</li>
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| | ==References== |
| <li>[http://www.forestry.crs.gov.ng/index.php/forum/5-laptop-discussion/225004-13102002-ish-on-sunday-13th-of-october-2002-new-balance-574-au.html#225933 http://www.forestry.crs.gov.ng/index.php/forum/5-laptop-discussion/225004-13102002-ish-on-sunday-13th-of-october-2002-new-balance-574-au.html#225933]</li>
| | * {{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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| <li>[http://www.aiyingfang.cn/bbs/showtopic-886222.aspx http://www.aiyingfang.cn/bbs/showtopic-886222.aspx]</li>
| | [[Category:Affine geometry]] |
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| </ul>
| |
In mathematics, an affine combination of vectors x1, ..., xn is a vector
called a linear combination of x1, ..., xn, in which the sum of the coefficients is 1, thus:
Here the vectors are elements of a given vector space V over a field K, and the coefficients are scalars in K.
This concept is important, for example, in Euclidean geometry.
The act of taking an affine combination commutes 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
Related combinations
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Affine geometry
References
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