Yeah, it’s funny how most of high school is spent focusing on the “exceptional” solvable cases of nonlinear equations (low degree polynomials, simple trig equations, exponentials) that one can come out with the skewed idea that solvability in nonlinear equations is more common than it actually is. While I understand that it’s important to build up a vocabulary of basic functions (along with confidence manipulating them) I think it is also important to temper expectations with the reality that nonlinear behaviors are so diverse and common that it is an a small miracle that we have somehow discovered enough examples of analytically solvable systems to enable us to understand a rich subset of behaviors!

One of my favorite perspectives on the difficulty of formulating a general theory of PDE in light of the difficulties posed by nonlinearities is Sergiu Klainerman’s “PDE as a unified subject” https://web.math.princeton.edu/~seri/homepage/papers/telaviv.... If I understand correctly, any general theory of PDE would have to incorporate all the subtle behaviors of nonlinear equations such as turbulence (which has thus far evaded a unified description). Indeed, ”solvable” nonlinear systems in physics are so special Wikipedia has a list of them https://en.m.wikipedia.org/wiki/Integrable_system. With this perspective, I’m tempted to say (in a non-precise manner) that solvable systems are the vanishingly small exception to the rule in a frighteningly deep sea of unsolvable equations.

Huh, somehow I also learned LA best through Griffiths QM. I almost wish Griffiths would write a Linear Algebra textbook.

Isn't it so good?! The way he starts the class having you thinking you missed a reading or something, and then by the end of the lecture you have an aha moment. Feels like he crams an aha moment into each lesson too, so good.

I agree here. I tried to power through the book (which to be fair is really clear, didactic and still concise) and I was stumped because I'm more practical minded, and I'd be trying to look for examples in my domain (signal processing: filtering, beamforming, mle/map, compression, etc.) and I'd be stumped quite fast.

I appreciate any textbook with interesting real world examples and slow worked-through solutions. But maybe I'm a bit too lazy to do the work myself.

I still haven't grasped really what svd does, why it is different from eigenstuff (and... well... what eigenvalues/vectors are...) and the link between those and solving linear systems, and with the characteristic polynomial, and matrix inversion, and... I have intuitions, and I can mostly implement the stuff, but no clear understanding.

So... I suck at learning linear algebra :-)