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What are your particular critiques about this book? There's nothing inherently inferior about a free resource. What makes LADR superior to this book? Finding a

What are your particular critiques about this book? There's nothing inherently inferior about a free resource. What makes LADR superior to this book? Finding a teacher isn't at all mandatory, you can do math yourself by completing exercises, checking your work, and looking at the solution afterwards. There are plenty of places to ask questions if you get stuck.

devereaux 8y ago
> What makes LADR superior to this book? Axler's approach for determinants.
- bollu 8y ago
  The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
  - Koshkin 8y ago
    > linear functions from 2^n to 3^n Say what?
    - nsajko 8y ago
      2^n is {0, 1}^n, the space of n-tuples with elements from the smallest field. The field is also known as GF(2), Z/2Z, or F₂.
  - ajkjk 8y ago
    A first-time student in linear algebra definitely does not need to hear anything about finite fields, though. I don't think anyone outside of certain mathematics subfields needs to.
  - skrealing 8y ago
    Given a vector space V, form its top wedge power. This is one-dimensional. And endomorphism of V induces an endomorphism of the top wedge power. But an endomorphism of a 1-dimensional vector space is a number. This number is the determinant. This is probably the correct generalization of the volume definition.
  - devereaux 8y ago
    Geometric interpretations have a place in learning: until one is ready to understand another interpretation that is more complete, better, yet more complicated. You don't start kids with complex numbers until they can handle reals. You don't even start negative numbers until they can handle positive numbers.
- skh 8y ago
  People who are going to be users of applied linear algebra would probably benefit from a traditional treatment of determinants. Axler's book is excellent for people who plan to go for more pure mathematics in their studies.
  - ajkjk 8y ago
    I would not wish the traditional treatment of determinants on anyone.
  - romwell 8y ago
    >benefit from a traditional treatment of determinants The only beneficial traditional treatment I can think of is saying "determinant is volume", and that's what Axler does. From there, one can look into alternating forms, convince oneself that an alternating n-linear form does the same thing, and obtain a formula for it (e.g. summing up signed products of numbers on the diagonals over all permutations of columns). If by "traditional" you mean: "Here's an insanely complicated formula that does something magical. Learn to compute it. On page 5, we'll prove that it tells something about independence. Oh, and we'll mention volume on page 10 briefly" -- then I not only think this is not useful, I think it's outright harmful.
- The_suffocated 8y ago
  Are you serious? Axler defines determinant as (up to a sign) the constant term of the characteristic polynomial, and he needs two different definitions for characteristic polynomial, one over R and one over C. Now what if the ground field is something else? Do we need yet another definition of characteristic polynomial in order to define the determinant? What if you are doing linear algebra over a commutative ring? LADR actually presents a very narrow view about linear algebra : it treats linear algebra merely as finite-dimensional functional analysis. The readers can be hit hard when they need to do other (computational or theoretical) stuffs. Similar concerns had been voiced on the internet before. In particular, I think Darij Grinberg's comments (below the answer https://mathoverflow.net/a/16996 https://mathoverflow.net/a/16996) on LADR are rather spot on. It's fine if you find LADR helpful. The book does have its merits (I like its clear and fluent writing and its neat proofs), but it has also its own shares of problems and there are other nice choices of books in the wild.
  - fsloth 8y ago
    Thanks. I found LADR not that helpful - I use linear algebra constantly but only on a very elementary level. After reading your message I now feel it's ok not to find that book applicable to my pursuits and that I'm not missing some concept I failed to grasp from the book.
  - hopler 8y ago
    The secret is that math is too big to fit into any one book. It takes many books with many perspectives you learn the generality and applicability of the topic. The average person who couldn't learn a topic read one book on the topic. The average mathematician read 5 or 10.
    - yesenadam 8y ago
      Good point. Even in maths, "bias" is inevitable. Every mathematician has a point of view. There's no objectively best curriculum or way of doing things, understanding things, solving problems etc. (I'm more familiar with this phenomenon in philosophy, where the greater the philosopher, the more they have entirely their own way of looking at things, untranslatable into another tongue, which you just have to come to understand on its own terms. A summary of their views leaves out the personal aspect, the style, the way of thinking, and will seem dead.)
romwell 8y ago
>Finding a teacher isn't at all mandatory I firmly disagree. What a teacher can give you is perspective that you, not knowing the subject, cannot have -- and augmenting any particular viewpoint expressed in a book. If people could learn everything from texts, we wouldn't have universities (for students) and conferences (for working mathematicians). In theory, one can write a Great Text that explains an Idea. In practice, it's damn hard to do that, and it's far easier to impart understanding in a conversation, filling in any blank spots the audience might have on the spot, and guiding the way in the jungle. That's why all texts are kind of bad. Either they are too narrow to give a wide perspective, or too huge to be absorbed! As one of my advisors said: mathematics, like food, is best shared. Don't go into it alone; and whether you have or don't have a mentor, try to find someone else to join you on your journey (a friend who wants to learn the same subject).

devereaux 8y ago

> What makes LADR superior to this book? Axler's approach for determinants.

bollu 8y ago
The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
- Koshkin 8y ago
  > linear functions from 2^n to 3^n Say what?
  - nsajko 8y ago
    2^n is {0, 1}^n, the space of n-tuples with elements from the smallest field. The field is also known as GF(2), Z/2Z, or F₂.
- ajkjk 8y ago
  A first-time student in linear algebra definitely does not need to hear anything about finite fields, though. I don't think anyone outside of certain mathematics subfields needs to.
- skrealing 8y ago
  Given a vector space V, form its top wedge power. This is one-dimensional. And endomorphism of V induces an endomorphism of the top wedge power. But an endomorphism of a 1-dimensional vector space is a number. This number is the determinant. This is probably the correct generalization of the volume definition.
- devereaux 8y ago
  Geometric interpretations have a place in learning: until one is ready to understand another interpretation that is more complete, better, yet more complicated. You don't start kids with complex numbers until they can handle reals. You don't even start negative numbers until they can handle positive numbers.
skh 8y ago
People who are going to be users of applied linear algebra would probably benefit from a traditional treatment of determinants. Axler's book is excellent for people who plan to go for more pure mathematics in their studies.
- ajkjk 8y ago
  I would not wish the traditional treatment of determinants on anyone.
- romwell 8y ago
  >benefit from a traditional treatment of determinants The only beneficial traditional treatment I can think of is saying "determinant is volume", and that's what Axler does. From there, one can look into alternating forms, convince oneself that an alternating n-linear form does the same thing, and obtain a formula for it (e.g. summing up signed products of numbers on the diagonals over all permutations of columns). If by "traditional" you mean: "Here's an insanely complicated formula that does something magical. Learn to compute it. On page 5, we'll prove that it tells something about independence. Oh, and we'll mention volume on page 10 briefly" -- then I not only think this is not useful, I think it's outright harmful.
The_suffocated 8y ago
Are you serious? Axler defines determinant as (up to a sign) the constant term of the characteristic polynomial, and he needs two different definitions for characteristic polynomial, one over R and one over C. Now what if the ground field is something else? Do we need yet another definition of characteristic polynomial in order to define the determinant? What if you are doing linear algebra over a commutative ring? LADR actually presents a very narrow view about linear algebra : it treats linear algebra merely as finite-dimensional functional analysis. The readers can be hit hard when they need to do other (computational or theoretical) stuffs. Similar concerns had been voiced on the internet before. In particular, I think Darij Grinberg's comments (below the answer https://mathoverflow.net/a/16996 https://mathoverflow.net/a/16996) on LADR are rather spot on. It's fine if you find LADR helpful. The book does have its merits (I like its clear and fluent writing and its neat proofs), but it has also its own shares of problems and there are other nice choices of books in the wild.
- fsloth 8y ago
  Thanks. I found LADR not that helpful - I use linear algebra constantly but only on a very elementary level. After reading your message I now feel it's ok not to find that book applicable to my pursuits and that I'm not missing some concept I failed to grasp from the book.
- hopler 8y ago
  The secret is that math is too big to fit into any one book. It takes many books with many perspectives you learn the generality and applicability of the topic. The average person who couldn't learn a topic read one book on the topic. The average mathematician read 5 or 10.
  - yesenadam 8y ago
    Good point. Even in maths, "bias" is inevitable. Every mathematician has a point of view. There's no objectively best curriculum or way of doing things, understanding things, solving problems etc. (I'm more familiar with this phenomenon in philosophy, where the greater the philosopher, the more they have entirely their own way of looking at things, untranslatable into another tongue, which you just have to come to understand on its own terms. A summary of their views leaves out the personal aspect, the style, the way of thinking, and will seem dead.)

romwell 8y ago

>Finding a teacher isn't at all mandatory I firmly disagree. What a teacher can give you is perspective that you, not knowing the subject, cannot have -- and augmenting any particular viewpoint expressed in a book. If people could learn everything from texts, we wouldn't have universities (for students) and conferences (for working mathematicians). In theory, one can write a Great Text that explains an Idea. In practice, it's damn hard to do that, and it's far easier to impart understanding in a conversation, filling in any blank spots the audience might have on the spot, and guiding the way in the jungle. That's why all texts are kind of bad. Either they are too narrow to give a wide perspective, or too huge to be absorbed! As one of my advisors said: mathematics, like food, is best shared. Don't go into it alone; and whether you have or don't have a mentor, try to find someone else to join you on your journey (a friend who wants to learn the same subject).