Affine space

In mathematics, an affine space may be defined somewhat abstractly as a set on which a vector space acts transitively.

Albeit somewhat jocular, the following characterization may be easier to understand: an affine space is what is left of a vector space after you've forgotten which point is the origin. 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, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space.