What's the difference between $A\in\mathbb{R}^{n\times n}$ and $A\in\mathcal{M}_{n\times n}(\mathbb{R})$ since no one asked yet because it might be a dumb question but I frequently encounter these two notations along in the same proof are they different from each other or do they both refer to the space of $n\times n$ matrices with real entries?
 A: I've only ever seen these notations refer to the same set. The latter is slightly less compact than the former, but the former could conceivably be misinterpreted as $\Bbb{R}^{n^2}$. Without any further context, I'd say they mean the same thing.
A: Notation varies a lot in math, and the exact meaning of what something means depends on the context.
$A\in\mathcal{M}_{n\times n}(\mathbb{R})$ means that $A$ is an element of the ring of $n \times n$ matrices with coefficients in $\mathbb{R}$.
$A\in\mathbb{R}^{n\times n}$ pretty clearly indicates that $A$ is an element of an $n \times n$ tuple with real coefficients.
These are not the same thing. With $A\in\mathcal{M}_{n\times n}(\mathbb{R})$ we know that $A$ is an element of a ring and so if we are given a $B\in\mathcal{M}_{n\times n}(\mathbb{R})$ we know how to compute $A+B$ and $AB$.
I feel like this is not is clear for $A\in\mathbb{R}^{n\times n}$. Like, if we are given $B\in\mathbb{R}^{n\times n}$, then perhaps you could compute $A+B$ and $AB$ by adding and multiplying the corresponding tuple entries. In this case, multiplying elements of $\mathcal{M}_{n\times n}(\mathbb{R})$ is very different than multiplying elements of $\mathbb{R}^{n\times n}$.
However, under the natural definitions of scalar multiplication and vector addiction, $\mathbb{R}^{n\times n}$ becomes an $n \times n$ dimensional vector space with real coefficients. Since $\mathcal{M}_{n\times n}(\mathbb{R})$ is also an $n \times n$ vector space with real coefficients, in this case we have that $\mathbb{R}^{n\times n}$ and $\mathcal{M}_{n\times n}(\mathbb{R})$ are isomorphic as vector spaces.
Sorry if this was a bit rambly, hope it helped.
