Affine Space

Let be a Vector Space over a Field , and let be a nonempty Set. Now define addition for any Vector and element subject to the conditions

1. ,

2. ,

3. For any , there Exists a unique Vector such that .

Here, , . Note that (1) is implied by (2) and (3). Then is an affine space and is called the Coefficient Field.

In an affine space, it is possible to fix a point and coordinate axis such that every point in the Space can be represented as an -tuple of its coordinates. Every ordered pair of points and in an affine space is then associated with a Vector .

See also Affine Complex Plane, Affine Connection, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity