Let $\mathbb K$ be a field and $n>1$ an integer. Then the $n$-fold cartesian product $\mathbb K^n$ can be seen both as a vector space and as an affine space.
As far as I know, one of the main theoretical difference between $\mathbb K^n$ as a vector space and $\mathbb K^n$ as an affine space has to do with the automorphism group $\textrm{Aut}(\mathbb K^n)$ of $\mathbb K^n$. In fact, I've heard that:
$i)$ When $\mathbb K^n$ is seen as a vector space then $\textrm{Aut}(\mathbb K^n)=\textrm{Gl}_n(\mathbb K)$. In particular, $\textrm{Aut}(\mathbb K^n)$ is a vector of dimension $n^2$.
$ii)$ When $\mathbb K^n$ is seen as an affine space then \begin{align*} \textrm{Aut}(\mathbb K^n)=\textrm{Gl}_n(\mathbb K^n) + \textrm{"translations"}. \end{align*} In particular, in this case, $\textrm{Aut}(\mathbb K^n)$ is a vector space of dimensionl $n(n+1)$.
Do you know any reference which discuss the above aspects of $\textrm{Aut}(\mathbb K^n)$?