Affine combination


In mathematics, an affine combination of is a linear combination
such that
Here, can be elements of a vector space over a field, and the coefficients 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 are elements of , 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
In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of, so the set of fixed points of forms an affine 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.

Related combinations