Comment by jacobolus | Hacker News Reader

Comment by jacobolus | Hacker News Reader

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Really? I find the part about the SVD in Axler's book extremely unhelpful, big blobs of opaque formulas and jargon with next to no explanation or context that basically require either knowing the topic fully beforehand or a huge amount of effort to parse.

e.g. Axler's definition of singular values is the extremely dry and technical:

> Suppose T is in L(V, W). The singular values of T are the nonnegative square roots of the eigenvalues of T†T, listed in decreasing order, each included as any times as the dimension of the corresponding eigenspace of T†T.

(Using a dagger instead of an asterisk for the conjugate transpose since HN interprets and asterisk to mean italics.)

If you already just proved a lot of stuff about eigenvalues, this could be a serviceable definition; at any rate it saves space. But it doesn't really explain the point.

I'd recommend anyone interested in this or related topics read Trefethen & Bau (1997) Numerical Linear Algebra.

From the preface

“You cannot read mathematics the way you read a novel.

If you zip through a page in less than an hour, you are probably going too fast.

When you encounter the phrase “as you should verify”, you should indeed do the verification, which will usually require some writing on your part.

When steps are left out, you need to supply the missing pieces.

You should ponder and internalize each definition.

For each theorem, you should seek examples to show why each hypothesis is necessary.”

It is a math studying book, not a let me chew it for you before you take it in type of a book. It requires effort, focus and missing steps are missing on purpose so you will discover them. This is the fun about studying mathematics.

Sorry you fell that way, but the book in my opinion is a masterpiece in math composition.

That's all well and good, but you also shouldn't excuse authors for not providing context, motivation, or explanation under the theory that the students should really be figuring out the whole subject from scratch for themselves.

If you are really lucky the result of not explaining things might be an occasional exceptional student who works out a correct personal concept. But more commonly the result is just an unfilled gap in understanding and either a moderately motivated student who develops fluency with symbol twiddling but doesn't get the point of what they're doing or a less-motivated student who decides the topic sucks and gives up.

I've read a lot of linear algebra books, and I personally find Axler's to have average quality exposition and a not tremendously insightful point of view. I know other people who swear by the book though, so YMMV. It's more appropriate for a well prepared pure math student who wants to go to grad school and has a goal of internalizing a lot of jargon so they can read/write pure math papers than for a scientist or computer programmer.

I agree completely. Sure, students could fill all the gaps themselves. But why not just fill the gaps and make good exercises?

They do. You get some in the prose, and then more in the exercises.

Learn ____ the hard way.

Or the book could fill in more details with a rich exposition and supplement with high quality exercises.

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.

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.

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.

Here you sit, an undergraduate with a calculus book open before you, or a pre-thesis graduate student with one of those books whose first ten pages, at least, you would like to master, or a research mathematician (established or would-be) with an article fresh off the press—what do you do now? How do you study, how do you penetrate the darkness, how do you learn something?

All I can tell you for sure is what I do, but I do suspect that the same sort of thing works for everyone. It's been said before and often, but it cannot be overemphasized: study actively. Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

— Paul Halmos, “I Want to Be a Mathematician”, Study.

Some books simply built around this protocol, some are making it “easier” so they’re more popular and sell more.

So you admit that chapter 7 does not read almost as poetry?

That is the poetry. When all parts clicks together.

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