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,

**and**

*a***are to be added. Jones draws an arrow from**

*b***to**

*p***and another arrow from**

*a***to**

*p***, and completes the parallelogram to find what Jones thinks is**

*b***+**

*a***, but is actually**

*b***+ (**

*p***−**

*a***) + (**

*p***−**

*b***). Similarly, Jones and Smith may evaluate any linear combination of**

*p***and**

*a***, or of any finite set of vectors, and will generally get different answers. However -- and note this well:**

*b*

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

See also affine geometry.