Vector space of $m\times n$ matrix $\Bbb R^{m\times n}$ vs vector space of a vector of size $mn$ $\Bbb R^{mn}$? I have a very basic question in linear algebra. Every vector of size $n$ with each entry from $\Bbb R$ lies in the space $\Bbb R^n$, where $\Bbb R^n$ is the cartesian product of $n$ copies of the set $\Bbb R$.
But how is it different from the vector space of matrices? It is written in my books that the vector space of matrices is $\Bbb R^{m\times n}$, but what does that mean? What is $\Bbb R^{m\times n}$? Because if it is the same as $\Bbb R^{mn}$, then how is an $m\times n$ matrix different from an $mn$ dimensional vector?
 A: Both spaces are isomorphic. For instance take $\phi:M_{m,n}(\Bbb R) \rightarrow \Bbb R^{mn}$ the morphism such that for $i_1\in\{1, ..., n\}$ and $j_1\in\{1, ..., m\}$ we have $\phi([\delta_{i, i_1}\delta_{j, j_1}]_{i, j})$ is the vector where all coordinates are 0 except the $i_1m + j_1$ -eth which is 1. Such a morphism exists since $[\delta_{i, i_1}\delta_{j, j_1}]_{i, j}$ is a basis of $M_{m,n}(\Bbb R)$.
You'll find that $\ker\phi = \{0\}$ and since $\dim M_{m,n}(\Bbb R) = \dim \Bbb R^{mn} = mn$, $\phi$ is bijective.
Thus a matrix in $M_{m,n}(\Bbb R)$ may be viewed as a vector in $\Bbb R^{mn}$.
Simply put, a matrix of $M_{m,n}(\Bbb R)$ is never anything more than $nm$ numbers, and neither is a vector in $\Bbb R^{mn}$.
A: Both spaces are isomorphic as vector spaces. In fact, often we will "flatten" matrices by converting them to $1\times(NM)$ vectors in just the way you are talking about. If the only thing you care about is the vector space addition and scalar multiplication, then there is no difference.
Where there is a difference is that the space of matrices has with it some extra structure. For instance, if you have $N = M$, meaning that you have square matrices, then you can multiply the matrices together to get a new matrix. This turns the space of $N \times N$ matrices into an algebra, which is a vector space with an additional way to "multiply" vectors together. Of course, the space of "flattened" vectors can also be turned into an algebra as well, so there is no real material difference, but it's important to note that when we talk about matrices we are implicitly including this extra structure as well.
