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In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.
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 (or, in the words of mathematical physicist John Baez, "An affine
space is a vector space that's forgotten its 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 Smith knows that it 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.
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