Affine combination

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

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

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

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

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 affine combinations commute 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 $T$ 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.