Reading about canonical forms for real matrices, I have found this:
Let $A$ be an $n\times n$ real matrix and let $T=T_A:\mathbb{R}^{n\times 1} \to \mathbb{R}^{n\times 1} $ be the associated linear operator.
I understand that $\mathbb{R}^{n\times 1}$ is the space of $1-$column matrices with $n$ rows, of the form $[x_1,x_2,\cdots,x_n]^T$, that usually, I identify with the vectors of $\mathbb{R}^n$.
What is the utility of distinguish between $\mathbb{R}^n$ and $\mathbb{R}^{n\times 1} $?
With this notation, every "space" being considered is of the form $\Bbb R^{m\times n}$. Otherwise, the author has to separately define $\Bbb R^n$.
I think most people like this distinction because we treat the underlying vector space $\Bbb R^n$ in a fundamentally different way than we treat the matrix space $\Bbb R^{m\times n}$. Obviously, this author didn't find that to be a useful distinction to make.
As the comments mention, there's also the matter of distinguishing row/column vectors.