Afterword

A course like this makes two kinds of promise, and it is worth saying at the end whether they were kept.

The first is the forward reference. Whenever a notebook said we come back to this — the tolerance that decides numerical rank, the conditioning that explains which least-squares method to trust, the theorem behind low-rank adaptation — a later notebook owed the reader delivery. Those promises are not kept by good intentions; they are kept by a ledger. tools/promises.yml records each one, and a gate in continuous integration fails the build if a promise is made and never met. If you found a forward reference in these pages that goes nowhere, it is a bug, and I would like to hear about it.

The second promise is the larger one: that every claim in this course is checked against something the calculation did not assume. Not “the code runs”. Not “the figure looks plausible”. Checked — against an exact algebraic identity, a closed form, an independent method, or a theorem with a sharp constant. There are several hundred such checks across the forty-seven notebooks, they all run on every commit, and a red one stops publication. That discipline is the reason I think this course is worth your time rather than one of the many other introductions to the same material.

It has a cost, and the cost is instructive. Designing an exercise whose answer can be verified independently is much harder than designing one whose answer you merely know. Several times while writing this I had a perfectly good exercise that could not be checked, and the rule is that such an exercise gets redesigned until it can be, or cut. What survives is material where the mathematics is sharp enough to be caught being wrong. That is a narrower course than one that allows itself to gesture, and I think it is a better one.

A word about what is missing. There is no multigrid, no domain decomposition, no serious treatment of parallel or communication-avoiding algorithms, and only a glance at structured eigenvalue problems. Interior-point methods and convex optimization proper are absent; so is the whole of numerical analysis for differential equations, which lives in the companion physics course. Tensor networks stop at matrix product states and never reach PEPS or DMRG proper. Each of those is a book. Naming them is the honest alternative to pretending the subject ends where the notebooks do.

And a word about the last volume. When I planned this course, Volume VIII was going to be two notebooks on regression and a nod at neural networks. It grew to seven because the request that shaped it was right: the linear algebra inside these systems deserves to be taught properly, at depth, with the checks attached, by somebody willing to say plainly which parts are linear algebra and which parts are not. §8.7 exists because the second half of that sentence matters as much as the first.

The subject is four hundred years old, the algorithms mostly seventy, and the applications in the last volume mostly seven. The mathematics did not change. It just kept turning out to be what everyone needed.

Raymond Amador Zürich, 2026