Affine combination

In mathematics, a linear combination of vectors x₁, ..., x_n

is called an affine combination of x₁, ..., x_n when the sum of the coefficients is 1, that is,

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 affine combinations commute with any affine transformation T in the sense that

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).

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