Affine spaces over a real vector space and displacement vectors

Let $V$ be a real vector space, with addition $+$, zero $0$, and additive inverses (Vector space over a field). An affine space over $V$ is a nonempty set $A$ together with a map

$+:A\times V⟶A,(p,v)⟼p+v,$

such that

1. $(p+v)+w=p+(v+w)$ and $p+0=p$ for all $p\in A$ and $v,w\in V$; and
2. for all $p,q\in A$ there is exactly one $v\in V$ with $p+v=q$.

The map in clause 1 is the action of $V$ on $A$. The unique vector supplied by clause 2 is written $q-p$ and is called the displacement from $p$ to $q$; thus $q-p$ is the unique $v\in V$ with $p+v=q$. An affine space over the real vector space $V$ is also called a real affine space modelled on $V$, and then $V$ is its space of displacements.

The following identities are consequences of the two clauses, with the displayed justification:

• $p+(q-p)=q$ for all $p,q\in A$: this restates the defining property of $q-p$ in clause 2.
• $p-p=0$ for all $p\in A$: taking $q=p$ in the previous display gives $p+(p-p)=p=p+0$, and clause 2 applied to the pair $(p,p)$ gives $p-p=0$.
• $p+0=p$ for all $p\in A$: this is clause 1.
• $(q-p)+(r-q)=r-p$ for all $p,q,r\in A$: $p+[(q-p)+(r-q)]=[p+(q-p)]+(r-q)=q+(r-q)=r=p+(r-p)$, so clause 2 applied to the pair $(p,r)$ identifies the two vectors.
• For each $p\in A$ the map $V\to A$, $v\mapsto p+v$, is a bijection: for every $q\in A$, clause 2 supplies $v=q-p$ with $p+v=q$, proving surjectivity; it is injective because $p+v=p+w$ gives $p+(v-w)=[p+v]+(-w)=p+w-w=p=p+0$, so clause 2 applied to $(p,p)$ gives $v-w=0$, that is, $v=w$.

Affine coordinate systems. Suppose $V$ is $4$-dimensional and $E:{\mathbb{R}}^4\to V$ is a linear isomorphism (Invertible linear maps, linear isomorphisms, and inverse linear maps). For a chosen point $o\in A$, define

$x(p):=E^{-1}(p-o),p\in A\mathrm{.}$

This is a bijection $x:A\to {\mathbb{R}}^4$: it is injective and surjective because $E^{-1}$ is a bijection and $p\mapsto p-o$ is a bijection of $A$ with $V$ by the bijection just proved, applied with that $o$. Such maps are the affine coordinate systems on $A$; clause 2 is exactly what makes them well defined, because it is clause 2 that attaches to each ordered pair $(o,p)$ a unique displacement $p-o$.