Affine combination

In mathematics, a linear combination of vectors $x 1, ..., x n$

\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 $x 1, ..., x n$ 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

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.

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Affine combination

See also

Related combinations

Affine geometry

References

External links