Affine combination

In mathematics, an affine combination of $x_{1}, ..., x_{n}$ 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, $x_{1}, ..., x_{n}$ can be elements (vectors) of a vector space over a field $K$, and the coefficients $$\alpha_{i}$$ are elements of $K$.

The elements $x_{1}, ..., x_{n}$ 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 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 $A$.

Related combinations

 * Convex combination
 * Conical combination
 * Linear combination

Affine geometry

 * Affine space
 * Affine geometry
 * Affine hull