[srean]

I have mixed feelings about this. BTW my comment is not about Axler's linear algebra book but math pedagogy in general.

Spoon feeding doesn't really help with learning, internalising, building internalisation for. Effort and frustration is key.

For me the best way has been to be forced to discover the important properties and results myself. It is slow but effective and best for knowledge retention. I think the best way to enforce is exercises with grades of progressive hints.

One of my best teachers used to ask us to prove things that are not true. The effort and frustration to prove those were the best teaching moments. We soon caught on to his method though and we're on our alert whenever we were asked to prove something to be true.

[jacobolus]

The big problem I have with Axler's presentation of the SVD is that it's backwards. It leads with a bunch of completely dry and technical minutiae written formally, loads students up with tedious and confusing technical exercises they aren't likely to appreciate, and defers the motivation, context, explanation, and pictures until a few dozen pages later (probably multiple weeks later for a course), and in my opinion doesn't do a great job with them even then. I think this does a big disservice to students, and I wouldn't recommend anyone learn about the SVD this way. (Admittedly, the SVD is treated more like a curious aside than a centrally important tool in Axler's book; he's not trying to train people to use numerical methods.)

The best place to learn about the SVD is probably in the context of some kind of concrete problem (the most illuminating would be a problem in statistics, image processing, geometry modeling, or whatever, but it could also be a more abstract pure math problem, something about quadratic forms or something) that demonstrates an actual need for it, and then introduce the idea with an intuitive explanation supported by pictures and spatial reasoning. The actual technical details are not really that complicated or hard to figure out once you understand the concept, but if you don't understand the concept then trying to prove a bunch of obscure technical statements just seems like pointless busywork.

[srean]

I mostly agree.

I wouldn't have liked Axler as my first book of linear algebra but as a second book, it worked, it gave me a more algebraic appreciation which is a bit against my natural grain of vector space and geometric way of thinking.

I quite liked Linear Algebra Done Wrong. My first linear algebra book was Noble's.