Affine combination

In mathematics, an affine combination of $x 1, ..., x n$ is a linear combination

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

such that

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

Here, $x 1, ..., x n$ can be elements (vectors) of a vector space over a field $K$ , and the coefficients $\alpha _{{i}}$ are elements of $K$ .

The elements $x 1, ..., x n$ can also be points of a Euclidean space, and, more generally, of an affine space over a field $K$ . In this case the $\alpha _{{i}}$ are elements of $K$ (or $\mathbb {R}$ for a Euclidean space), and the affine combination is also a point. See Affine space § Affine combinations and barycenter for the definition in this case.

This concept is fundamental in Euclidean geometry and affine geometry, as the set of all affine combinations of a set of points form the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span.

The affine combinations commute 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 $A$ .

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.

Affine combination

See also

Related combinations

Affine geometry

References

External links