Affine combination

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

$\sum_{i=1}^{n}{\alpha_{i} \cdot x_{i}} = \alpha_{1} x_{1} + \alpha_{2} x_{2} + \cdots +\alpha_{n} x_{n}$

in which the sum of the coefficients is 1, thus:

$\sum_{i=1}^{n} {\alpha_{i}}=1$ .

Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients $α i$ are scalars in K.

This concept is important, for example, in Euclidean geometry.

Motivation

Imagine that Smith knows that a certain point is the origin, and Jones believes that another point -- call it p -- is the origin. Two vectors, a and b are to be added. Jones draws an arrow from p to a and another arrow from p to b, and completes the parallelogram to find what Jones thinks is a + b, but is actually p + (a − p) + (b − p). Similarly, Jones and Smith may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However -- and note this well:

If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!

The proof is a routine exercise. Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure" -- i.e., the values of affine combinations.

There is no number other than 1 with which the same idea works.

Affine combination

Motivation

See also