Franklin's Notes

Affine map

An affine map $A:X\to Y$ between affine spaces (or between convex subsets thereof) is a function that preserves affine combinations : whenever $\lambda+\mu=1$. If $A$ is an affine map, then it also preserves affine combinations of more than $2$ vectors: for any such affine combination can be rewritten as a sequence of binary affine combinations by recursively applying the following identity:

If there exists another affine map $B:Y\to X$ which is an inverse of $A$, i.e. such that $BA:X\to X$ and $AB:Y\to Y$ are the identity maps, then we say that $A$ is an affine isomorphism of $X$ and $Y$. If $X=Y$, then we say that $A$ is an affine automorphism.

Question 1. If $A:X\to Y$ is an affine map that is a set-bijection between the respective points of $X$ and $Y$, then is it necessarily an affine isomorphism?



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