Affine combination

Template:Short description In mathematics, an affine combination of x₁, ..., x_n is a linear combination

$\sum _{i=1}^n{\alpha }_i\cdot x_i={\alpha }_1{x}_1+{\alpha }_2{x}_2+\cdots +{\alpha }_n{x}_n,$

such that

$\sum _{i=1}^n{\alpha }_i=1.$

Here, x₁, ..., x_n can be elements (vectors) of a vector space over a field K, and the coefficients ${\alpha }_i$ are elements of K.

The elements x₁, ..., x_n can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the ${\alpha }_i$ are elements of K (or $\mathbb{ℝ}$ for a Euclidean space), and the affine combination is also a point. See Template:Slink 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 T in the sense that

$T\sum _{i=1}^n{\alpha }_i\cdot x_i=\sum _{i=1}^n{\alpha }_i\cdot T{x}_i.$

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, Template:Mvar, acts on a column vector, Template:Vec, the result is a column vector whose entries are affine combinations of Template:Vec with coefficients from the rows in Template:Mvar.

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.