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 $(A,V,+)$, where $A$ is a set, $V$ is a Vector Space (over $\mathbb{K}$) and $+:A\times V\to A$ is a Group Action which is Regular (i.e. free and transitive) of the additive group $(V,+)$ of its underlying vector space $V$.

Transformations which preserve affine structures are called Affine Maps.

Properties

Vectors act as translations of points in the space

Vectors $\overrightarrow{ab}\in V$ act as “parallel displacements”: for each point $a\in A$, one can go to $b\in A$ via group action¹, as

$$b=a+\overrightarrow{ab}$$

There needs to be at least one such $\overrightarrow{ab}$ for each $a,b\in X$ due to transitivity, and there is only one such $\overrightarrow{ab}$ since the group action is free.

Affine spaces are $\mathbb{K}$-torsors

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

Fixing a point induces a vector space

Picking a fixed point $O\in X$, one can prove that Fixing a point in an affine space induces a vector space via the bijection $m_O:A\leftrightarrow V$. Through this function (and its inverse $m_O^{-1}$), one can induce vector space operations on $X$ as

$$\begin{array}{rl}x+y& :=m_O^{-1}(\overrightarrow{Ox}+\overrightarrow{Oy})\\ \lambda x& :=m_O^{-1}(\lambda \overrightarrow{Ox})\end{array}$$

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

Footnotes

1. Existence of this vector between any two points in $A^n$ is guaranteed by the transitive property, and being free guarantees it to be unique. ↩