Translations in affine spaces are affine automorphisms

up:: Affine Isomorphism

Let be an Affine Space, and let be the translation of all points in by (i.e. ). It defines an Affine Map with underlying linear map¹

such that one can see the map as

Note that this is the case, since the diagram commutes. Let be points connected by ─ i.e. . Then we have that

since the Group Action is “compatible” ─ ─, alongside the additive group being Abelian.

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

with underlying linear map

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References

Footnotes

1. Note that this works since, for any group action, ↩