Affine combination

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

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

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

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

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

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

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 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

• Convex combination
• Conical combination
• Linear combination

Affine geometry

• Affine space
• Affine geometry
• Affine hull

References

• Gallier, Jean (2001), Geometric Methods and Applications, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95044-0. See chapter 2.

External links

• Notes on affine combinations.