In mathematics, an affine combination of 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, can be elements (vectors) of a vector space over a field , and the coefficients <math>\alpha_{i}</math> 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 <math>\alpha_{i}</math> are elements of (or <math>\mathbb R</math> 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

:<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, , 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

  • Convex combination
  • Conical combination
  • Linear combination

Affine geometry

  • Affine space
  • Affine geometry
  • Affine hull

References

  • . See chapter 2.
  • Notes on affine combinations.