Affine Space

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“An affine space is like a vector space that has forgotten its origin.” (John Baez)

An affine space consists of a triple , where is a set, is a Vector Space (over ) and is a Group Action which is Regular (i.e. free and transitive) of the additive group of its underlying vector space .

Transformations which preserve affine structures are called Affine Maps.

Properties

Vectors act as translations of points in the space

Vectors act as “parallel displacements”: for each point , one can go to via group action, as

There needs to be at least one such for each due to transitivity, and there is only one such since the group action is free.

Affine spaces are -torsors

Therefore, one can see the “difference between points” in the affine space as being vectors in . In fact, an affine space is an example of a -Torsor (seeing as the additive group ).

Fixing a point induces a vector space

Picking a fixed point , one can prove that Fixing a point in an affine space induces a vector space via the bijection . Through this function (and its inverse ), one can induce vector space operations on as

A consequence of this is that An affine space with a fixed point is isomorphic to its underlying vector space.

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References

• torsors (John Baez)
• ARNOL’D, Vladimir Igorevich. Mathematical methods of classical mechanics. Springer Science & Business Media, 2013.
• Notes on Mathematical Physics for Mathematicians - Daniel Tausk
• Affine spaces | Mauricio Poppe