Affine combination

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In mathematics, an affine combination of vectors x1, ..., xn is a vector

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

called a linear combination of x1, ..., xn, in which the sum of the coefficients is 1, thus:

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

Here the vectors are elements of a given vector space V over a field K, and the coefficients ${\displaystyle \alpha _{i}}$ are scalars in K.

This concept is important, for example, in Euclidean geometry.

The act of taking an affine combination commutes 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 subspace (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.