An affine map is an affine isomorphism iff its underlying linear map is an isomorphism

up:: Affine Isomorphism

Transclude of Affine-Map.excalidraw

Let and be Affine Spaces¹. Let be an Affine Isomorphism with underlying linear map .

Affine map is bijective Underlying map is isomorphism

Let be an affine isomorphism (Affine Map which is a Bijection). Then, for any , we have

Since has an inverse, we have that

Since the inverse is also an affine map, we have that

where is the underlying linear map associated to .

However, note that, since an affine space is determined by a Regular Group Action, then there is one and only one vector connecting and , which is . Thus,

And, thus, the underlying linear map of an affine isomorphism is a Vector Space Isomorphism.

Affine map is not bijective Underlying map is not an isomorphism

Let not be an affine isomorphism; then it is not Injective or it’s not Surjective (or neither).

Suppose is not injective. Then there are , in which case but . Thus, is not an isomorphism, since it does not take an origin to an origin (i.e. it’s not injective).

Suppose is not surjective. Then there are points in which are not reached by .
Let . Then there are points in which have a distance of, say, from and which are still in . ?????

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References

• Notes on Mathematical Physics for Mathematicians - Daniel Tausk

Footnotes

1. is just a vector space (possibly) unrelated to . Not to be confused with ‘s dual. ↩