Let $\mathrm{M}(n,\mathbb{R})$ be the space of $n\times n$ matrices over $\mathbb{R}$. Consider the function
$m \in \mathrm{M}(n,\mathbb{R}) \mapsto (a_{11},\dots,a_{1n},a_{21}\dots,a_{nn}) \in \mathbb{R}^{n^2}$.
The space $\mathrm{M}(n,\mathbb{R})$ is locally Euclidean at any point and we have a single chart atlas. I read that the function is bicontinuous, but what is the topology on $\mathrm{M}(n,\mathbb{R})$?
Second question... in what sense it is defined a $C^{\infty}$ structure when there are no (non-trivial) coordinate changes? Do we have to consider just the identity change?
Thanks.