Affine combination: Encyclopedia II - Affine combination - Motivation

Affine combination - Motivation

Suppose points in space are associated with vectors. Imagine that Smith knows that a certain point is the origin, and Jones believes that another point -- call it p -- is the origin. Two points, 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 a point that 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:

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.

Adapted from the Wikipedia article "Motivation", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki