I am trying to understand properly what a 'computation' is, from a mathematical perspective. I'm interested in understanding both the formalism, and also how to 'think' about intuitively. Reading various discussions across the stack exchange sites, I have at least inferred that a computation seems to be a very general 'thing'. However I also couldn't find a question on math stack exchange discussing exactly what a computation is, or how it should be thought of.
At least from the wikipedia article on computation, it's very generally defined as a 'calculation', which is itself defined as a 'deliberate process that transforms one or more inputs into one or more results'. I found this definition interesting, because to me this seems very much like it can be characterised mathematically as a map, or at least a relation between sets. Is it useful to rigorously define a computation this way?
It also got me thinking about what computation is/means with regards to the above definition, with a particular example from linear algebra. Very early on, we're introduced to the fact that linear maps can be characterised in the form of matrices. For example, given a linear map $T: \mathbb{F}^n \rightarrow \mathbb{F}^m$ and the standard basis in $\mathbb{F}^n$, we have:
$$T(\textbf{x}) = \sum_{i=1}^n x_k\textbf{a}_k$$
where $\textbf{x} = (x_1,...x_k)$, and $\textbf{a}_k = T(\textbf{e}_1)$. I think this is an awesome result; we have that the system $\{\textbf{a}_k\}_{k=1}^n$ spans the image of the linear map, and it gives us a way to fully characterise the morphism with just $n$ vectors in the target space. The next step, at least in the textbook I use, does what I initially thought was entirely trivial: we express the above using matrices. We define the matrix multiplication $\textbf{A}\textbf{x}$ exactly so that it IS $\sum_{i=1}^n x_k\textbf{a}_k$. The only thing we do is realise that there's a nice way to 'compute' this, because the column by coordinate rule becomes apparent. And of course, we create this new matrix object $\textbf{A}$, by combining information of the system of vectors above.
It seems to me that mathematically, there really is no difference between whether we think about is as $T(\textbf{x}) = \sum_{i=1}^n x_k\textbf{a}_k$, or $ T(\textbf{x})=\textbf{A}\textbf{x}$. It seems like both contain the exact same information. Albeit the matrix is a slightly different object to the system of vectors, but this seems like an utterly technical distinction.
Now my question arises: is there anything mathematically meaningful in using matrices to represent linear transformations rather than as sums of scalars? A big part of my curiosity is that somehow it seems to have huge implications for physical computing, and yet there seems to be no 'conceptual' difference between the two. Further, are both the matrix itself, and the linear map $T$ considered computations in their own right?
I am not quite sure if this question is appropriate for MSE, because I'm not sure if this question is more suitable for physics, or computer science, or posed as a mathematical question. I unfortunately don't have enough knowledge of any of these to know if what I'm talking about is a reasonable maths question (or if it even makes sense). Originally I wanted to understand what a computation was in the physical sense, but realised there was some discussion of it on MSE. I'd greatly appreciate pointers to Q&A on other stack exchange sites, or recommendations for introductory reading to the topic from a mathematical perspective.