By reading the first sentence in this article I interpret that, for every projective space, every isomorphism of its underlying vector space gives rise to an isomorphism of projective spaces.
Is this also true for affine spaces (and their underlying vector space)?
Or, at the very least, is every isomorphism of affine spaces induced by an isomorphism of the underlying vector space?
Let $\mathcal A$ and $\mathcal A'$ be affine spaces over the vector spaces $E$ and $E'$ respectively, both over the same field.
If
$$ \varphi \;:\; E \;\longrightarrow \; E' $$
is a linear isomorphism, then the function $ \; f \;:\; \mathcal A \;\longrightarrow \; \mathcal A' \;$ defined by $$ f(P) = O'+\varphi(P-O), $$
where $O$ and $O'$ are arbitrarily fixed points of $\mathcal A$ and $\mathcal A'$ respectively, is an affine isomorphism. In fact, \begin{align} f(P) - f(Q) &= \big(O'+\varphi(P-O)\big) - \big(O'+\varphi(Q-O)\big) =\\[1.5ex] &= \varphi(P-O) - \varphi(Q-O) =\\[1.5ex] &= \varphi(P-Q). \end{align}
The isomorphism $f$ depends on the arbitrary choice of O and O', so it is not unique.
The vice versa is obvious precisely because of the definition of isomorphism of affine spaces:
Def. $\;$ A function $\;f\;:\; \mathcal A \to \mathcal A'\;$ is an affine isomorphism iff there is a linear isomorphism $\varphi\;:E\;\to E'$ such that
$$ f(P)-f(Q) = \varphi(P-Q)\qquad \forall \;P, Q\in \mathcal A. $$
In this case, $\varphi$ is unique since $P-Q$ describes all $E$, so that $\varphi$ is known over its entire domain.