I have seen diverse methods of defining the underlying variety of the linear algebraic groups $GL_n$ and $PGL_n$. I was then asking myself, if we could define them the following way: $GL_n = \mathrm{Spec\ Aut}(\mathbb{A}^n)$ and $PGL_{n+1}=\mathrm{Spec\ Aut}(\mathbb{P}^n)$.
This seems to easy to be true and I have seen nobody do it.