Translations in affine spaces are affine automorphisms

up:: Affine Isomorphism

Let $(E,V,{+}_E)$ be an Affine Space, and let $T_w:E\to E$ be the translation of all points in $E$ by $w\in V$ (i.e. $T_w(P)=P+w$). It defines an Affine Map with underlying linear map¹

$$\begin{array}{rl}L:V& \to V\\ v& \mapsto v\end{array}$$

such that one can see the map as

$$T_w(P+v)=T_w(P)+L(v)=T_w(P)+v$$

Note that this is the case, since the diagram commutes. Let $P,Q$ be points connected by $v$ ─ i.e. $Q=P+v$. Then we have that

$$T_w(Q)=Q+w=(P+v)+w=\cdots =T_w(P)+v=T_w(P)+L(v)$$

since the Group Action is “compatible” ─ $(A+v)+w=A+(v+w)$ ─, alongside the additive group $(V,+)$ being Abelian.

Thus, this is an Affine Isomorphism, since its inverse is

$$\begin{array}{rl}{T}_w^{-1}:E& \to E\\ P& \mapsto P+(-w)\end{array}$$

with underlying linear map

$$\begin{array}{rl}{L}^{-1}:V& \to V\\ v& \mapsto v\end{array}$$

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References

Footnotes

1. Note that this works since, for any group action, $Q=P+v⟺P=Q+(-v)$ ↩