I imagine that's the right way to do it. Treating a linear map as a table of numbers is too concrete, and there are far too many conventions for how to do it. A

I imagine that's the right way to do it. Treating a linear map as a table of numbers is too concrete, and there are far too many conventions for how to do it. Abstractly, it's a type-(1,1) tensor, which is naturally isomorphic to a linear map. (I'm slightly abusing how I use the term "naturally isomorphic"). There are many notations for tensors that don't mention indices, including the somewhat surprising Abstract Index Notation by Roger Penrose. For the sake of completion: A vector is a type-(1,0) tensor. A linear form (or "covector" if you prefer this term) is a type-(0,1) tensor. A bilinear form is a type-(0,2) tensor. The point of saying the type of a tensor, is to clarify what geometric object an array of numbers represents. A curse (and a blessing) of matrix notation is that it doesn't tell you this. Both a type-(0,2) and type-(1,1) tensor can be represented by matrices, but the first is used for (let's say) dot products, and the second is used for linear transformations. The only problem with type-(m,n) notation is remembering whether "m" is the contravariant order or the covariant order. It turns out that it's contravariant, which is the opposite to what the naming would suggest. In programming, we would do {cov: m, con: n}, so we wouldn't have to remember.