[dan-robertson]

So the point of this article is that differentiation is linear. That is, the operator D which takes f to d f/d x is linear. The author points out that one can write this down in as a matrix with respect to a basis of polynomials, which is nice for suitably well behaved functions and I think nice for understanding. Other operators one might look at are integration, Fourier or Laplace transforms, or more exotic integral transforms which are linear. One can view a Fourier transform like a change of basis.

In another sense, derivatives themselves are linear: for a function f: U -> V of vector spaces, the derivative (at some point) is a linear map from U -> V, (i.e. the derivative of the functions is a function Df: U -> L(U,V)) and this extends the concept of derivative to multiple dimensions as f(x+h) = f(x) + (Df)(x)h + o(h).

This seems ok at first derivatives but can become unwieldy as they became tensors higher rank.

Another question one might ask on learning that differntiation is a linear operator is what it’s eigenvalues are. For differentiation these are functions of the form f(x) = exp(ax). But one can construct other linear operators and from this you get Sturm–Liouville theory which is fantastic.

One final note is that much of this multidimensional derivatives and tensor stuff becomes a lot easier if one learns suffix notation (aka Einstein notation, aka index notation, aka summation convention), as well as perhaps a few identities with the kronecker delta or Levi-Civita symbol. Notation can break down a bit with arbitrary rank tensors: $a_{i_1,...,i_k}$ becomes unwieldy but writing $a_{pq...r}$ is ok.