Affine combination explained

In mathematics, an affine combination of is a linear combination

\sum ^{n} _{i =1} {\alpha _{i} ⋅ x _{i} } = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that \sum ^{n} _{i =1} {\alpha _{i} }=1.

Here, can be elements (vectors) of a vector space over a field, and the coefficients

\alpha _{i} 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

\alpha _{i} are elements of (or R 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

T\sum ^{n} _{i =1} {\alpha _{i} ⋅ x _{i} } = \sum_^. In particular, any affine combination of the fixed points of a given affine transformation T is also a fixed point of T , so the set of fixed points of T 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

Related combinations

• Convex combination
• Conical combination
• Linear combination

Affine geometry

• Affine space
• Affine geometry
• Affine hull

References

• . See chapter 2.

External links

• Notes on affine combinations.