Affine combination

Affine combination

In mathematics, an affine combination of vectors "x"1, ..., "x""n" is vector : sum_{i=1}^{n}{alpha_{i} cdot x_{i = alpha_{1} x_{1} + alpha_{2} x_{2} + cdots +alpha_{n} x_{n}, called the linear combination of "x"1, ..., "x""n" , in which the sum of the coefficients is 1, thus:

:sum_{i=1}^{n} {alpha_{i=1 .

Here the vectors are elements of a given vector space "V" over a field "K", and the coefficients alpha _{i} are scalars in "K".

This concept is important, for example, in Euclidean geometry.

An affine combination of fixed points of an affine transformation is also a fixed point, so the fixed points form an affine subspace (in 3D: a line or a plane, and the trivial cases, a point and the whole space).

ee also

* affine space
* affine geometry
* affine hull
* convex combination
* conical combination

References

*. "See chapter 2".


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