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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 clar

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.

jjoonathan 3y ago
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.
- fiforpg 3y ago
  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.
  - jjoonathan 3y ago
    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.
- nerdponx 3y ago
  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.
CamperBob2 3y ago
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.
- ayhanfuat 3y ago
  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.
- noqc 3y ago
  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.
  - CamperBob2 3y ago
    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.
    - Koshkin 3y ago
noqc 3y ago
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."
senderista 3y ago
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.
Joker_vD 3y ago
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).

jjoonathan 3y ago

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.

fiforpg 3y ago
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.
- jjoonathan 3y ago
  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.
nerdponx 3y ago
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.

CamperBob2 3y ago

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.

ayhanfuat 3y ago
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.
noqc 3y ago
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.
- CamperBob2 3y ago
  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.
  - Koshkin 3y ago

noqc 3y ago

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."

senderista 3y ago

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.

Joker_vD 3y ago

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).