Every vector space induces an affine space

up:: Vector Space

Let $(V,+,\cdot )$ be a Vector Space. Then one can define an Affine Space $(E,V,+)$ in which $E=V$ and $(V,+)$ is its underlying vector space.

Define by the Group Action

$$\begin{array}{rl}\alpha :V\times (V,+)& \to V\\ (p,v)& \mapsto p+v\end{array}$$

Note that it is indeed a group action, since

• $\forall p\in V:p+0=p$
• $\forall p\in V,\forall u,v\in (V,+):(p+v)+w=p+(v+w)$

Note that this group action is Regular, since, for any two points $p,q\in V$, there is a unique vector connecting them — namely, $q-v\in (V,+)$, seeing $p,q\in (V,+)$.

Thus, $\left(V,(V,+),+\right)$ is an affine space induced by $(V,+,\cdot )$.