An affine space with a fixed point is isomorphic to its underlying vector space

up:: Affine Space

Given an affine space $(A,V,+)$, Fixing a point in an affine space induces a vector space. Fix a point $O\in A$, and call its induced Vector Space $A_O$.

Then one can easily construct a Vector Space Isomorphism $T:A_O\leftrightarrow V$, since all points have a unique vector associated to their translation from $O$ (since there is a Regular Group Action), and since all vectors have an associated point in $A$ which comes from $O$, since the Group Action’s domain is defined over the entirety of the additive group $(V,+)$ and is, per hypothesis, Free and Transitive ─ which ensures that there are no “lazy” vectors: all vectors “induce movement” upon the set $A$.

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References

• Affine transformation - Wikipedia