Real square matrices space as a manifold 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.

The topology on $\mathrm{M}(n, \mathbb{R})$ making the map you give bicontinuous is, well, the topology that makes that map bicontinuous. Less tautologically, it is the topology induced by any norm on $\mathrm{M}(n, \mathbb{R})$. (Recall that all norms on a finite-dimensional real vector space are equivalent.)
Because Euclidean space has a canonical smooth structure, the fact that you have a single-chart atlas means that you can give $\mathrm{M}(n, \mathbb{R})$ the smooth structure by pulling it back from $\mathbb{R}^{n^2}$. There are no compatibility conditions to verify: the identity transition map will always be smooth.
