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Comment by fiforpg | Hacker News Reader

About avoiding determinants to the degree that this book does: while I agree it makes sense to delay introducing them, the goal should not be avoidance but clarity. The way author has to bend himself backwards here when dealing with eigenvalues isn't great either.

I would recommend Strang for a healthy balance in handling determinants.

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Axler is pathological in his avoidance of determinants. I've heard (third hand) that he once pulled aside some fields medalist into a classroom after a talk and asked them "Do you like determinants?" I imagine him drawing the curtains and sweeping for bugs first.

I attended a (remote) seminar where he was talking about this book, and this seems more or less accurate. Mathematicians are a weird lot.

The response that he received in the story was "I feel about them the same way I feel about tomatoes. I like to eat them, but other than that, no, I don't like them."

I read Strang and then Axler. Strang is great at numerics but weak at presenting the abstract picture. I feel like if I had taken, say, finite elements (or any other subject where it's important to take the abstract / infinite dimensional picture seriously before reducing to finite dimensions) right after Strang without reading LADR then I'd have been seriously underprepared.

You have a point in that to understand any particular subject well, it makes sense to read more than one book on it, at least to compare the different perspectives.

Also worth noting that Strang has a couple of similar linear algebra books, so we might not even be discussing the same text.

That's entirely possible, but in the context of introductory books I think it's fair to assume & limit scope to Strang's "Introduction to Linear Algebra" and Axler's "Linear Algebra Done Right."

I am an applications-oriented person and my inclination was to go directly from a matrix/determinant heavy picture into applications. Strang['s intro text] only. I am extremely glad that someone intercepted me and made me get some practice with abstract vector spaces, operators, and inner product spaces first, using Axler. This practice bailed me out and differentiated me from peers on a number of occasions, so I want to pass down the recommendation.

FWIW I think this is the benefit of Strang. If you're in science or engineering or statistics, often you don't need the general picture, and IMO too much generality gets in the way of understanding. Start with a good understanding of the most important cases that appear in applied work, and drill them until you're fluent with them. Then generalizing will be easier.

Honestly, I think Strang is overrated. Yeah, I know, on HN that's like criticizing Lisp or advocating homebrew cryptography or disagreeing that trains fix everything. But still.

I bought his 6th ed. Introduction to Linear Algebra textbook, and he doesn't get more than two pages into the preface before digressing into an unjustified ramble about something called "column spaces" that appears in no other reference I've seen. (And no, boldfacing every second phrase in a math book just clutters the text, it doesn't justify or explain anything.) Leafing through the first few chapters, it doesn't seem to get any better.

The lecture notes by Terence Tao that someone else mentioned look excellent, in comparison.

His lectures are great but I definitely agree about the book. It reads like one of the TAs transcribed the lectures and added some exercises to the end.

In my experience, it's a little bit easier for new students to understand that the image of a matrix is the span of its columns, hence column space.

Perhaps, but that's about as useful as pointing out that monads are a monoid in the category of endofunctors. What's the "image of a matrix?" Coming at LA from a 3D graphics background, I've never heard that term before. And what does the "span of its columns" mean?

To me, each column represents a different dimension of the basis vector space, so the notion that X, Y, and Z might form independent "column spaces" of their own is unintuitive at best.

These are all questions that can be Googled, of course, but in the context of a coherent, progressive pedagogical approach, they shouldn't need to be asked. And they certainly don't belong in the first chapter of any introductory linear algebra text, much less the preface.

In the context of linear algebra, a matrix is a linear map. A map is characterized by its domain and its image. These are very important characteristics.

> To me, each column represents a different dimension of the basis vector space, so the notion that X, Y, and Z might form independent "column spaces" of their own is unintuitive at best.

I can't help but feel a treatment of linear algebra that assumes all matrices are invertible by default isn't a very good treatment at all. Column spaces are exactly how you harness your (very useful!) intuition that the columns of a matrix are where the basis goes. I agree that it should be defined before use, but it is- in the textbook proper. The preface is for the author to express themselves!

Now row spaces are an abomination, but that's because I'm not really a computation guy. I'm sure they're great if you get to know them.

I definitely covered the column space and row space in my undergrad LA class, long before I had ever heard of Strang.

An exceptional minority of people has the ability to learn linear algebra in its full abstract generality as their first treatment of the material, and come away with something resembling an understanding.

The rest of us dopey oafs must develop intuition carefully from specific concrete examples that extend gradually from algebra and geometry that we are familiar with already. Those of us in this sad deficient category must be led painstakingly over several weeks of course material to even the basic idea that a matrix is just a particular representation and special case of something called a linear transformation.

If you are one of the former type, you are blessed, but it's unfair to sneer at the latter, and it will only do your students a disservice.

I can guarantee you chatgpt can explain column and row spaces to you, suggesting that it is part of the common lexicon in linear algebra.

I'm not saying they don't belong in the book. I'm saying, evidently poorly, that they don't belong in the first chapter.

Strang has a bit of a "Next, draw the rest of the fucking owl" vibe going on. It wasn't what I'd been led to expect from the reviews.

In most of the LA courses determinants just... feel almost completely unmotivated, their definition just "comes down from the heavens in all its mysterious glory", and wow, how convenient that those things have all those nice properties!.. although they don't seem to actually be used for much unless your LA course actually contains elements of elimination theory which most of them don't, for some reason (even though it would seem to be quite a useful part of mathematical knowledge but apparently not).

Also, if you prefer an abstract approach, the determinant is just the nth exterior power of a linear transformation :) No need to introduce a basis at all, at least in principle.

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