Let $k$ be a field, let $V$ be a [vector space](/page/Vector%20Space) over $k$, and let $A$ be an [affine space](/page/Affine%20Space) modeled on $V$, meaning that $A$ is equipped with a free and transitive right action $A\times V\to A$, written $(p,v)\mapsto p+v$, of the additive group of $V$, satisfying $p+0=p$ and $(p+u)+v=p+(u+v)$ for all $p\in A$ and $u,v\in V$. Fix a point $o\in A$. Define the map $\phi_o:A\to V$ by declaring $\phi_o(p)=p-o$, where $p-o$ denotes the unique vector $w\in V$ such that $o+w=p$. Then $\phi_o$ is a bijection. Moreover, for every $p\in A$ and every $v\in V$, $\phi_o(p+v)=\phi_o(p)+v$.