# Affine combination

In mathematics, an **affine combination** of vectors *x*_{1}, ..., *x*_{n} is a vector

called a linear combination of *x*_{1}, ..., *x*_{n}, in which the sum of the coefficients is 1, thus:

Here the vectors are elements of a given vector space *V* over a field *K*, and the coefficients 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

In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of , so the set of fixed points of 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.

## See also

### Related combinations

### Affine geometry

## References

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