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Comment by artagnon | Hacker News Reader

I'm not too fond of the example-driven approach. The student will certainly be able to multiply simple matrices and compute determinants in some simple cases, but will not be able to answer fundamental questions like "what can I put in a matrix?", or "what is a vector field?" (the author gives 10 "rules" with no structure, so it's impossible for the student to memorize them). The author throws around `R` everywhere, but all his examples use `Z`. For a book on linear _algebra_, there is very little algebraic rigor in the book.

The standard textbook for learning linear algebra at the undergraduate level is Hoffman and Kunz, which is far better.

In short, this is a poor textbook.

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In my experience teaching my friends and colleagues there are two ends of the learning spectrum--you can learn by example first or by theory first, or both, but the former is by far the lion's share of the population.

I've taught people where there's absolutely no way I can teach them a generalization unless it's rote--however they can get domain-specific knowledge and beat me over the head in their domain and teach me something about my generalization as well that I might have been missing.

The best way for me to teach calculus to most people is to walk around a room explaining concepts like velocity, speed, acceleration, and position. However, I personally would never learn it this way--I prefer Spivak, during my read through I learned more than I could ever learn in a class.

It's also my belief that most people aren't like me in the sense that they learn from example first. However, most people I know are like me so it's hard to realize how needed a textbook like this is and how huge it could be for the regular person to learn something like this.

If you think about it, pretty much everyone learns math through example first. You learn arithmetic in elementary school, and it might only be after an additional decade or more of schooling that you actually learn the logical underpinnings which make the basic arithmetic operations work. If you actually started with boolean logic and tried to build up all of mathematics from there without at some point tying it back to a tangible example which can be understood in terms of the real world I'm not sure anyone could successfully learn it.

So I actually had a basic algebra system before they taught it to me, for what it's worth, probably something I figured out from the structure of the language my parents were using (both accountants). I'm a pretty rare individual, I've never had to work at math.

I took a long time to learn how to learn from example, it used to be impossible but now it's almost just as natural.

True. Examples are necessary for both motivation, and for showing how the formalisms work, at this level. One good undergraduate textbook that has plenty of examples is Dummit & Foote. However, that doesn't mean that you don't define anything properly, and solely use examples to show how to crunch certain computations. The student is then stuck with a bunch of patterns in their heads, with little mathematical understanding.

My first course in linear algebra was extremely formal with very little practical examples or visual intuition. It’s easy not to appreciate just how much intuition can go a long way in helping us understand a concept, even if the intuition is formally “fuzzy”. I used Hefferon’s Linear Algebra to plug the gaps in my intuitive and practical knowledge, and was all the better for it.

People have different reasons and approaches for learning, and in the linear algebra landscape Hefferon is a worthy addition.

(edit: just like one of the commenters below I bought the hard copy a few years ago to show my support and thanks for the free edition.)

How is "what can I put in a matrix?" a fundamental question? Looking at the textbook, the author certainly covers the fundamental questions in vector spaces, linear maps, and decomposition/normal forms.

There's even a really great section on projective geometry that would have been helpful before seeing it quickly defined as homogenous coordinates before moving on to the definition as a quotient manifold of a sphere and Lie group.

Are you complaining about the axioms for a vector space? Because that list will exist for any LA class lol. That's like complaining that a college Calculus class is teaching the limit definition of the derivative.

I think (from only having skimmed the contents) that there is some handwaving around the definition of the characteristic polynomial. In order to properly define it, you need to allow matrices over arbitrary rings, not just fields.

Aren't these the standard vector space axioms? I seem to recall that they were defined in a quite similar way in my class.

Of course, you can simply the definition if you can say "vector spaces form a group and also satisfy these axioms with respect to scalar multiplication", but then you need to introduce groups first, which would be a distraction.

Thanks for your perspective. Not sure why you got the downvotes. What do you think of "Linear Algebra Done Right"?

I think it's the leap from - 'I'm not a fan of the approach' to 'this is a poor textbook'. Just because they don't like the approach doesn't mean it won't be effective for other learners. Some people absolutely need examples before they can attach anything theoretical to a concept.

Yes, the wording in that comment is not very friendly.

Also, I think that examples motivate people. I'm sure a lot of people are capable of understanding highly abstract things, but motivation is key.

Interesting approach. He starts off with vector spaces, after providing some intuition, and uses it to drive the rest of the book. He defines matrices in terms of linear maps, and talks about operations on vector spaces, with small sections dedicated to the corresponding matrix computations. There's no hand-waving or lack of rigor.

Overall, a fine textbook.

There's also the Terse Introduction to Linear Algebra, which is shorter and more rigorous than Linear Algebra Done Right. All of these are known texts which have generated some body of discussion.

An idiosyncrasy of Axler's book is that it introduces determinants only toward the end.

> An idiosyncrasy of Axler's book is that it introduces determinants only toward the end

I think the idea is that determinants do not necessarily belong in linear algebra proper, at least not with as prominent a position as they are usually awarded in textbooks. More specifically the idea is that determinants are being excessively and detrimentally used in linear algebra proofs. Axler actually published a paper called "Down With Determinants!" in 1995, for which he won the Lester R. Ford Award. Quote from the introduction:

> This paper focuses on showing that determinants should be banished from much of the theoretical part of linear algebra. Determinants are also useless in the computational part of linear algebra.

Paper: https://www.maa.org/sites/default/files/pdf/awards/Axler-For...

Wikipedia page for the award: https://en.wikipedia.org/wiki/Paul_R._Halmos_%E2%80%93_Leste...

That seems like a personal crusade. Of course, determinants are computationally mostly irrelevant, and of course, they actually belong to multilinear algebra, but theoretically, determinants are quite nice and they are useful in many LA proofs, especially when it comes to the characteristic polynomial.

You are being dismissive of Axler (with the "personal crusade" comment) and disagree with his position, but it would be less disrespectful of Axler and more beneficial to the discussion (and interesting to me) to give at least some argumentation to match Axler's. That is, why is Axler's approach (in the short paper or the book) worse than the traditional approach?

It's not. For all I care he may write his book without introducing determinants or only introducing them at the very end. I'm all for different pedagogical approaches.

But he writes a paper called "down with determinants" and starts it with "if you think complex matrices have an eigenvalue because the characteristic polynomial has a root, then this is wrong". But it's not wrong, mathematically it's 100% correct! It's just his own personal opinion because he somehow doesn't like determinants.

Mathematics is all about different approaches and different tools, not about "the one true enlightened way", as he makes it out to be. This is why I'm calling it a "personal crusade". Had he just called his paper "a determinant-free approach to linear algebra" I would take no issue.

From my skimming of that article, Axler's main complaint with determinants is that it pedagogically leaves students with the impression that eigenvalues are somehow a property of the determinant rather than being a fundamental property of a linear transformation.

I'm not sure I've ever met anyone who has that view; in my experience, more people leave linear algebra with a laundry list of matrix properties whose utility isn't obvious, with determinants and eigenvalues both on that list. This in general is a pedagogical issue with teaching linear algebra as plug-and-chug techniques without explaining why we're doing them.

Truthfully, I'm not convinced that introducing the determinant as "just" the product of eigenvalues is itself useful. One of the more useful properties of the determinant is that it is built up via row-reduction, and consequently, it can be computed using Gaussian elimination. I've seen some texts actually define the determinant via its row-reduction properties, and then build up to demonstrating the other formulas for it.

I'll also point out that there are two more uses that Axler doesn't acknowledge: the determinant tells you the volume of a parallelpiped, and it's a convenient way to remember the formula for a cross product.

The degree to which classrooms and textbooks have converged onto Axler's pedagogical approach is the degree to which Axler's position is "personal" vs mainstream.

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