Difference between revisions of "Affine combination"

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==See also==
 
==See also==
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===Related combinations===
 
===Related combinations===
 
{{details|Linear combination#Affine, conical, and convex combinations}}
 
{{details|Linear combination#Affine, conical, and convex combinations}}
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==References==
 
==References==
 
* {{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''.
 
* {{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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== External links ===
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[http://graphics.cs.ucdavis.edu/education/GraphicsNotes/GraphicsNotes/Affine-Combinations/Affine-Combinations.html Notes on affine combinations.]
  
 
[[Category:Affine geometry]]
 
[[Category:Affine geometry]]
 
[[he:צירוף אפיני]]
 
[[hu:Affin kombináció]]
 
[[nl:Affiene combinatie]]
 
[[pl:Kombinacja afiniczna]]
 
[[pt:Combinação afim]]
 
[[vi:Tổ hợp afin]]
 

Latest revision as of 19:32, 2 October 2014

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

Template:Rellink

Affine geometry

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}. See chapter 2.

External links =

Notes on affine combinations.