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.

- 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

Corollaries: 1 2

Definitions: 3 4 5 6 7 8 9 10 11 12 13 14 15

Proofs: 16 17

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