# Definition: Affine Space

An afine space $$\mathcal A$$ over a field $$F$$ is a triple $$\mathcal A:=(A,V_A,v)$$, in which

• $$A$$ is a set,
• the elements $$P\in A$$ are called points.
• $$V_A$$ is a vector space over the field $$F$$ ,
• $$v:A\times A\mapsto V_A$$ is a function with the following properties:
• Any two points $$P,Q\in \mathcal A$$ uniquely determine a vector $$x\in V_A$$ with $$x=v(P,Q)=P-Q=\overrightarrow{PQ}$$. We might also express it equivalently as $$P=Q+x$$, i.e. we get the point $$Q$$, if we translate the point $$P$$ by the vector $$x$$.
• For all $$P,Q,R\in \mathcal A$$, we have $$\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}$$, i.e. if we translate the point $$P$$ by one vector and then translate the translated vector by another vector, then the final point is a single translation of the initial point by a third vector.

Corollaries: 1 2
Definitions: 3 4 5 6 7 8 9 10 11 12 13 14 15
Proofs: 16 17

Github: ### References

#### Bibliography

1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994