Affine combination

In mathematics, an affine combination of Template:Math 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, Template:Math can be elements (vectors) of a vector space over a field Template:Math, and the coefficients $\alpha _{i}$ are elements of Template:Math.

The elements Template:Math can also be points of a Euclidean space, and, more generally, of an affine space over a field Template:Math. In this case the $\alpha _{i}$ are elements of Template:Math (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, as the set of all affine combinations of a set of points form the smallest subspace 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 Template:Math in the sense that

$T\sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\sum _{i=1}^{n}{\alpha _{i}\cdot Tx_{i}}.$

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 subspace (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

• Template:Citation. See chapter 2.

External links

• Notes on affine combinations.