Affine combination

In mathematics, an affine combination of is a linear combination
:$\sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_,$
such that
:$\sum_^ =1.$

Here, can be elements ( vectors) of a vector space over a field , and the coefficients $\alpha_$ are elements of .

The elements can also be points of a Euclidean space, and, more generally, of an affine space over a field . In this case the $\alpha_$ are elements of (or $\mathbb R$ for a Euclidean space), and the affine combination is also a point. See 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 in the sense that
:$T\sum_^ = \sum_^.$
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, , acts on a column vector, , the result is a column vector whose entries are affine combinations of with coefficients from the rows in .

See also

Related combinations

* Convex combination
* Conical combination
*Linear combination

Affine geometry

* Affine space
* Affine geometry
* Affine hull

References

* {{Citation , last1=Gallier , first1=Jean , authorlink=Jean Gallier , title=Geometric Methods and Applications , publisher=Springer-Verlag , location=Berlin, New York , isbn=978-0-387-95044-0 , year=2001 , url-access=registration , url=https://archive.org/details/geometricmethods0000gall . ''See chapter 2''.

External links

Notes on affine combinations.

Affine geometry