Affine combination
Template:Short description In mathematics, an affine combination of Template:Math is a linear combination
- <math> \sum_{i=1}^{n}{\alpha_{i} \cdot x_{i}} = \alpha_{1} x_{1} + \alpha_{2} x_{2} + \cdots +\alpha_{n} x_{n}, </math>
such that
- <math>\sum_{i=1}^{n} {\alpha_{i}}=1. </math>
Here, Template:Math can be elements (vectors) of a vector space over a field Template:Math, and the coefficients <math>\alpha_{i}</math> are elements of Template:Math.
The elements Template:Math can also be points of a Euclidean space, and, more generally, of an affine space over a field Template:Math. In this case the <math>\alpha_{i}</math> are elements of Template:Math (or <math>\mathbb R</math> for a Euclidean space), and the affine combination is also a point. See Template:Slink 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 Template:Math 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 points 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 space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).
When a stochastic matrix, Template:Mvar, acts on a column vector, Template:Vec, the result is a column vector whose entries are affine combinations of Template:Vec with coefficients from the rows in Template:Mvar.
See alsoEdit
Related combinationsEdit
Affine geometryEdit
ReferencesEdit
- Template:Citation. See chapter 2.