Affine combination

In mathematics, an affine combination of vectors x1, ..., xn 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, ..., xn, 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

Affine geometry

References

External links