Affine combination
In mathematics, an affine combination of x1, ..., xn is a linear combination
- [math]\displaystyle{ \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]\displaystyle{ \sum_{i=1}^{n} {\alpha_{i}}=1. }[/math]
Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients [math]\displaystyle{ \alpha_{i} }[/math] are elements of K.
The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the [math]\displaystyle{ \alpha_{i} }[/math] are elements of K (or [math]\displaystyle{ \mathbb R }[/math] for a Euclidean space), and the affine combination is also a point. See Affine space § Affine combinations and barycenter 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 T in the sense that
- [math]\displaystyle{ 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]\displaystyle{ T }[/math] is also a fixed point of [math]\displaystyle{ T }[/math], so the set of fixed points of [math]\displaystyle{ 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, 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
Affine geometry
References
- Gallier, Jean (2001), Geometric Methods and Applications, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95044-0, https://archive.org/details/geometricmethods0000gall. See chapter 2.
External links
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