I am just beginning to learn about Lie groups and am made somewhat uncomfortable by the textbook's handwavy decision to talk about Lie groups $GL(V)$ where $V$ is some $n$-dimensional real vector space. It is not clear to me that when when $V\cong\mathbb R^2$ and $End(V)$ is made isomorphic to $\mathbb R^4$ after a choice of basis for $V$, that that $\mathbb R^4$ cannot be exotic (as I know nothing about exotic things except their existence). I believe that maybe the requirement that matrix multiplication be differentiable might settle the issue, but I have no idea to go about showing this.
Otherwise, if I choose to believe that all $n$-dimensional vector spaces for $n\neq 4$ have a unique differentiable structure (they have unique topologies for sure), it becomes obvious that talking about $GL(V)$ for arbitrary real vector spaces is perfectly sensical.
So in short, can $GL(\mathbb R^2)$ ever be a Lie group not diffeomorphic to standard $\mathbb R^4$?