Affine space ↩

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created: 2022-01-13 12:08:32
modified: 2022-01-13 12:18:45

An affine space is a set together with a vector space and the action of on the set . Explicitly:

That has the following properties:

  1. Right identity: , where is the zero vector in .
  2. Associativity: , : (where the last is the addition in )
  3. Free and transitive action: For every , the mapping : is a bijection.
  4. Existence of one-to-one translations: For all , the mapping : is a bijection.

The first two properties are simply defining the properties of a (right) group action. The elements of the affine space are called points.