Affine combination

In mathematics, an affine combination of x₁, ..., x_n is a linear combination

${\displaystyle \sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n},}$

such that

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

Here, x₁, ..., x_n can be elements (vectors) of a vector space over a field K, and the coefficients ${\displaystyle \alpha _{i}}$ 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 ${\displaystyle \alpha _{i}}$ are elements of K (or ${\displaystyle \mathbb {R} }$ 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

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

In particular, any affine combination of the fixed points of a given affine transformation ${\displaystyle T}$ is also a fixed point of ${\displaystyle T}$, so the set of fixed points of ${\displaystyle T}$ 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.