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Comment by btilly | Hacker News Reader

The first mistake that I found is, All eigenvectors that correspond to one eigenvalue lie on the same line, but have different magnitudes.

The eigenvectors for [[1 0] [0 1]] are 2 dimensional.

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Can you give an example of two eigenvectors that do not lie on the same line? (I am curious, I am not a mathematician)

What he gives is the identity matrix—multiply on the right by any 2x1 vector and you will get that vector back. So any such vector, provided it is not the zero vector, is an eigenvector with eigenvalue 1.

[1 0] and [0 1] are both eigenvectors of the identity matrix, yet not on the same line.

I highly recommend http://www.axler.net/DwD.html for developing a good intuition about what eigenvalues and eigenvectors actually are.

I obviously formulated this wrong. Will rephrase, but doesn't the intuiton generally hold that when you find an eigenvector for one eigenvalue, then you can construct infinite number of eigenvectors by scaling that one?

Yes, that is trivially true. In fact any linear combination of eigenvectors for the same eigenvalue will be an eigenvector for that eigenvalue.

Another important point to appreciate is that eigenvalues and eigenvectors frequently require complex numbers. For example the eigenvalues of [[0 1] [-1 0]] are i and -i. This geometrically correlates with some kind of "rotation".

The problem arises when there is degeneration (some eigenvalues are repeated). See for example https://www.ch.ntu.edu.tw/~byjin/course/ChemMath/handouts/de...

[edit: the rest of my original comment was not correct]

If an eigenvalue is unique, the corresponding eigenvectors are on a line. If it appears twice, the corresponding eigenvectors are [edit: could be] on a plane. In general, they span [edit: could span] a subspace of dimension equal to the multiplicity.

Not true. The matrix [[1 1] [0 1]] has the eigenvalue 1 repeated twice but the eigenvectors are on a line.

The generalized eigenvectors are on a plane.

The case where the eigenvectors do not span the space corresponds to matrices whose Jordan normal form has upper triangular bits.

The dimension of the space spanned by the eigenvectors associated with a degenerate eigenvalue is bounded above by the algebraic multiplicity of the eigenvalue and bounded below by one.

You seem to have reworded to "All eigenvectors that correspond to one a unique eigenvalue lie on the same line, but have different magnitudes." This is technically true if you interpret "unique" to mean "multiplicity of one", but still seems confusing, especially as you have not yet discussed the multiplicity of eigenvalues.

I would consider rephrasing the article as you do in the above comment, "when you find an eigenvector for one eigenvalue, then you can construct infinite number of eigenvectors by scaling that one".

The book based on this paper is one of the best text books I've read. I highly recommend reading it if you're looking for a solid understanding of linear algebra.

As someone who worked their way through Strang (we did a little study group and did the homework assigned to some class at Rutgers), I'd love it if someone could take a moment to explain what they like so much about Axler. I've tried a couple times to slog my way through Axler and had not much success.

I have both Axler and Strang. I learned from Axler but occasionally use Strang as a reference. I find that Axler emphasizes the algebra part of linear algebra while Strang emphasizes the linear part. One isn't better than the other but one might make more sense to different people.

Axler avoids determinants with a passion. For good reason, too, because they are not helpful in understanding how spaces interact. I forget how Strang proves the spectral theorem but compared to Lax the pedagogy could not be more different. Determinants are really nice tools for short proofs, but not helpful in understanding. But Axler is the way you want to think about linear algebra as a practitioner.

Axler is easier to digest if you have some familiarity with the tools and the jargon but hated/forgot everything in your college linear algebra course.

A (very imperfect) analogy might be something like the GoF book vs Peter Norvig's essay on design patterns.

I’m not in the target audience and have neither learned nor taught from Axler’s book, so I can’t tell you precisely what people like about it, but it is aimed at pure math undergraduates who are interested in using linear algebra in much more abstract contexts, and focuses more on proofs than many introductory linear algebra textbooks.

Not sure if they’re useful (I haven’t watched them), but Axler made a series of videos about the core content of his book https://www.youtube.com/watch?v=lkx2BJcnyxk&list=PLGAnmvB9m7...

it is aimed at pure math students who are interested in using linear algebra in much more abstract contexts

I don't think that's really true. The book itself puts it much more simply and directly, at the very beginning in 'Preface for the Instructor':

"You are about to teach a course that will probably give students their second exposure to linear algebra."

You don’t think this book is aimed at pure mathematics students?

It’s certainly not aimed at numerical analysis students, or engineering students, or physics students. (Which isn’t to say that those students can’t take pure math courses if they want.)

I mean in the context of Strang. I don't think Strang is aimed strictly at, say, engineering students. To me, the difference is the starting point, more than anything else.

As far as I can tell Strang’s target audience is something like: most undergraduates at MIT who didn’t already learn the subject before arrival, except the pure math students who are likely to substitute a more theoretical course (18.700 or 18.701 vs. 18.06).

Right. Each is aimed at somewhat different audiences. We're undoubtedly getting into some pretty fine hairsplitting, the thing I was whining about is 'pure maths students' and 'everyone else' is not an accurate way to describe them.

Not a contradiction. Their first exposure would have been something like multivariable Calculus. So students have seen matrices, but they were magic, unmotivated, and probably didn't make much sense except in a "here is the formula to memorize" kind of way.

Axler tries to teach students how to understand linear algebra.

If all that you want to do is use it, the prospect of that understanding may not be very motivating.

I didn't say there was some contradiction. I just don't think the key difference between Strang and Axler is really the degree of mathsiness.

I was commenting about what Axler is, and not comparing it to anything else.

My point is that the comment that you quoted from the preface in no way changes the fact that the book's point is to convey an understanding of linear algebra that is primarily of interest to people going on in math.

Now I happen to think it is the right way to understand linear algebra and is how people in other fields should think about it. Because it is easier to figure out again if you've not done it in a while. But this point of view is primarily going to motivate would be mathematicians.

and not comparing it to anything else.

Sure, but the topic is 'Given Strang, what's the deal with Axler'. It's a perfectly sensible question in its own right.

It is probably logically unnecessary but I think it's best to have experience with the "computational" aspects of linear algebra before diving into Axler's book.

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