Affine Space

An affine space is a set together with a vector space and a faithful and transitive group action of (with addition of vectors as group operation) on . In particular, being an abelian group, it turns out that the only vector acting with a fixpoint is (i.e., the action is simply transitive, hence both transitive and free, whence free) and there is a single orbit (the action is transitive). In other words, an affine space is a principal homogeneous space over the additive group of a vector space.

Explicitly, an affine space is a point set together with a map

with the following properties:

1. Left identity

2. Associativity

3. Uniqueness

   is a bijection.

The vector space is said to underlie the affine space and is also called the difference space.

By choosing an origin, one can thus identify with, hence turn into a vector space. Conversely, any vector space, is an affine space over itself. The uniqueness property ensures that subtraction of any two elements of is well defined, producing a vector of .

If, and are points in and is a scalar, then

is independent of . Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

By noting that one can define subtraction of points of an affine space as follows:

is the unique vector in such that ,

one can equivalently define an affine space as a point set, together with a vector space, and a subtraction map with the following properties :

1. there is a unique point such that and
2. .

These two properties are called Weyl's axioms.