Suppose you have a function $f$ that takes a single real number as input, and it outputs some vector in the form $\begin{bmatrix}a\\b\end{bmatrix}$, where $a$ and $b$ are both real numbers.. For example, you input $3$ and it outputs $\begin{bmatrix}1\\2\end{bmatrix}$. What does this look like in arrow notation?
I would suggest $f:\mathbb{R} \to \mathbb{R}^2$, but I'm not sure.
I think this is correct because the input space consists of one dimension, but the output space consists of two dimensions.
Or how about if it was the other way around, where you input $\begin{bmatrix}1\\2\end{bmatrix}$ and it outputs $3$. Would this be $f:\mathbb{R}^2 \to \mathbb{R}$?
And how does our approach to this kind of notation generalise? If the input space has $n$ dimensions, and the output space has $m$ dimensions, do we then say $f:\mathbb{R}^n \to \mathbb{R}^m$?
How about for matrices? If the input space consists of matrices of dimension $n\times m$, and the output space consists of matrices of dimension $a\times b$, do we then say $f:\mathbb{R}^{n\times m} \to \mathbb{R}^{a\times b}$?
Yes to all of them. For the last question about matrices, what you have written is acceptable. Another notation people often use if $M_{n \times m}(\Bbb{R})$ for the space of $n \times m$ (rows $\times$ columns) matrices with entries in $\Bbb{R}$. So, you might also see it written as \begin{equation} f: M_{n \times m}(\Bbb{R}) \to M_{a \times b}(\Bbb{R}) \end{equation}
So, in general if you're mapping from a set $A$ into a set $B$, we write $f:A \to B$.