I suppose this is part terminology question and part math, but I am trying to untangle what we mean when we say "The linear function $f$ maps $R_{m\times n}$ space to $R_{m}$ space", and "The linear function $f$ maps $R_{m\times n}$ space to $R_{1}$ space".
To wit:
Let us say that there exists an $m\times n$ matrix $A$, and a function $f$ that maps $R_{m\times n}$ space to $R_{m}$ space. In this case, such a function $f$ can be an $n\times 1$ vector $v$. Thus, to apply the function, we have simply:
$ f(A_{m\times n}) = A_{m\times n}v_{n\times 1} =b_{m\times 1}$
So here, the function $f$ is the vector $v$.
My question is when we read the statement "The linear function $f$ maps the $R_{m\times n}$ space to $R_{1}$ space", (scalar), then what is an example of this function? It cant be just a vector or just a matrix, so what does it look like? I realize we can do $v^{T}Av$ if $A$ is a square, and this will give a scalar, but what is the function here?
Thanks