What is an example of a linear function that maps a matrix to a scalar? What makes it a 'function'? 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
 A: A linear function from ${\mathbb R}^{m \times n}$ to $\mathbb R$ can be written in the form $f(A) = \text{Tr}(A B)$ where $B \in {\mathbb R}^{n \times m}$ and $\text{Tr}$ means trace. 
A: A function is just a set of rules for generating a single output (injective), given a certain input. You don't have to be able to write in down in an explicit way.
A classic example for a function that takes a matrix and outputs a scalar is the determinant, which is a perfectly good (and very useful) function.
A: If you remove the specification "linear" (as in the original formulation of the question), then there are many functions you might consider. A function to $\mathbb R$ is just a way of assigning a member of $\mathbb R$ (i.e. a Real Number) to each matrix.
This could be done at random, for example, but the functions of interest tend to be more regular. You could send each matrix to the number in the top left-hand corner, for example (normally position 1,1) - actually that function, and the others which send the matrix to the numbers in other positions pretty much underlie all that we do with matrices: so much so that we forget the functions exist.
You can make other functions out of multiple elements - the sum of all the elements would be one, for example, or the sum or product of a particular row or column. Again there are functions which are more interesting because they encapsulate features of the matrix which allow analysis or classification. Particularly important are ones which are unchanged when the basis of the underlying space is changed. As well as the Determinant and Trace, there is the Rank, for example. (See comments below too)
