My understanding is that all linear operators on finite dimensional Banach spaces have matrix representations (is that correct?).
If so, is there in principle some algorithmic way of mapping (bounded) linear operators to the appropriate matrix? Or is there some subset of linear operators for which this is the case? If not, is there a nice way of seeing why this is not possible? Comments on the infinite dimensional case are also welcome.
If $X$ is a finite dimensional Banach space over $\mathbb{R}$, then $X$ is isomorphic to $\mathbb{R}^n$ (as a Banach space). This is just the fact that all norms on finite dimensional normed spaces are equivalent. In particular, this implies that if $X,Y$ are finite dimensional Banach spaces, then any linear map $f:X\rightarrow Y$ is continuous.
Suppose that $f:X\rightarrow Y$ is a linear map and $X,Y$ are finite dimensional. Then we can view $f$ as a linear map $\mathbb{R}^n\rightarrow \mathbb{R}^m$ for some $n,m\in \mathbb{N}$ by picking bases of $X$ and $Y$. Now you can pick the usual standard bases matrix representation of this linear map.