Affine combination

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 Affine space § Affine combinations and barycenter 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 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 space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).

When a stochastic matrix, A, acts on a column vector, b→, the result is a column vector whose entries are affine combinations of b→ with coefficients from the rows in A.