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".