Difference between $\mathbb{R}^{m \times n}$ and $\mathbb{R}^m \times \mathbb{R}^n$?

Question

Would a matrix of size $m \times n$ (with only real-valued entries) be an element of $\mathbb{R}^{m \times n}$ or $\mathbb{R}^m \times \mathbb{R}^n$?

Answer 1

Why $\mathbb{R}^m\times \mathbb{R}^n$ is Wrong

Simple counting will explain why an $m\times n$ matrix cannot be identified with an element of $\mathbb{R}^m \times \mathbb{R^n}$: a typical element of $\mathbb{R}^m \times \mathbb{R}^n$ is an ordered pair where the first element comes from $\mathbb{R}^m$ and the second element comes from $\mathbb{R}^n$. In turn, an element of $\mathbb{R}^k$ is an ordered tuple of elements from $\mathbb{R}$. So a typical element of $\mathbb{R}^m \times \mathbb{R}^n$ looks like $$ \renewcommand{\vec}[1]{\mathbf{#1}} (\vec{x}, \vec{y}) = ( (x_1, x_2, \dotsc, x_m), (y_1, y_2, \dotsc, y_n ) ). $$ This ordered pair can, in turn, be identified with a tuple with $m+n$ elements: $$ (\vec{x}, \vec{y} ) \sim ( x_1, x_2, \dotsc, x_m, y_1, y_2, \dotsc, y_n ).$$ This tuple is an element of $\mathbb{R}^{m+n}$, as it contains $m+n$ terms. But an $m\times n$ matrix contains $mn$ terms. Thus, unless $m=n=2$, an $m\times n$ matrix cannot be identified with an element of $\mathbb{R}^m \times \mathbb{R}^n$.

Is $\mathbb{R}^{m\times n}$ Right?

Suppose that $A = (a_{jk})$ is an $m\times n$ matrix with entries in $\mathbb{R}$. That is, $$ A = \begin{pmatrix} a_{11} & a_{12} & \dotsb & a_{1n} \\ a_{21} & a_{22} & \dotsb & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dotsb & a_{mn} \\ \end{pmatrix}. $$ As noted above, this matrix contains $mn$ terms. If we were to list out all of the terms as a tuple, then we could write something like $$ A \sim (a_{11}, \dotsc, a_{1n}, a_{21}, \dotsc, a_{2n}, \dotsc, a_{m1}, \dotsc, a_{mn} ) \in \mathbb{R}^{m\times n}. $$ From the point of view of counting elements, it is not wrong to suggest that $A \in \mathbb{R}^{m\times n}$.

That being said, I would use some care with that notation. The notation $\mathbb{R}^{m\times n}$ is often interpreted to mean the $(m\times n)$-fold Cartesian product of $\mathbb{R}$ with itself (as a set), or to mean the vector space consisting of real vectors of length $mn$. There certainly are books which use $\mathbb{R}^{m\times n}$ to denote the space of $m\times n$ matrices, so, again, it is not wrong to use this notation.

However, if you want to avoid ambiguity, I would suggest that the notation $$ M_{m\times n}(\mathbb{R}) $$ is a common notation which is much less likely to be misunderstood.