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> Why should anyone read this book if it presents the subject in a wrong way? What is particularly done “wrong” in the book? Does he ever answer this question?

> Why should anyone read this book if it presents the subject in a wrong way? What is particularly done “wrong” in the book? Does he ever answer this question? It's posed at the start of the book, but then immediately ignored.

Syzygies 2y ago
The title is a jab at and a response to "Linear Algebra Done Right" (https://linear.axler.net https://linear.axler.net) by Sheldon Axler. Sheldon's book is more suitable for a second course in linear algebra. It's an excellent book, that my more advanced students have enjoyed. Sheldon, however, approaches the subject with some rigid dogma: Determinants are evil. His view is they're not practical to compute at scale. It can be good to learn theory with one's determinant hand tied behind one's back, just as it can be good to learn "constructive" mathematics, but this should not be one's only approach. I used to attend an academic sponsor's day at the math institute MSRI (now SLMath). Afternoons reached to keep us busy. One year Sheldon's wife demonstrated some academic teaching software, and my group was tasked to solve a linear algebra problem that Sheldon had provided. Unaware that everyone was watching us on the big screen, I solved it instantly, using determinants. I still don't understand Sheldon's longer solution.
- munchler 2y ago
  This is very informative. Thank you for the explanation.
- btilly 2y ago
  His view is not just that they are not practical to compute at scale. (They actually are practical to compute. Just not with the formula. Use row reduction instead.) His view is that they are presented as an arbitrary piece of magic that people really don't understand. And, exactly because they don't really understand the determinant, they wind up not properly understanding fundamental concepts in linear algebra. Such as Jordan normal form. If your goal is actually to properly understand linear algebra, he's absolutely right. If your goal is to use linear algebra for a wide variety of practical purposes, though, determinants are often essential.
  - xanderlewis 2y ago
    Since a significant amount of my knowledge/intuition about linear algebra comes from his book, I now feel not only that I don’t like determinants but that I’m allergic even to matrices. Everything seems so much nicer and friendlier if you think of linear maps between finite-dim vector spaces instead. When I hear, for example, ‘upper triangular matrix’ I have to translate it in my head into something like ‘matrix representing a linear operator that preserves the standard flag’ in order to actually feel like I understand it. Of course, I’m not a programmer or an engineer of any kind, so I have the luxury of not needing the computational efficiency.
    - staunton 2y ago
      What's a "standard flag"? Couldn't find anything about it... I would think "upper triangular" would be a weird/bad notion to talk about in a basis-independent setting (because it depends on the basis).
      - xanderlewis 2y ago
        A ‘flag’ is just a funny name for an increasing (and therefore increasing in dimension) sequence of subspaces. The ‘standard flag’ is the sequence spanned by the standard basis vectors: start with {0}, then things of the form (x,0,…), then (x,y,0,…), and so on. * What I’m saying is that an upper triangular matrix preserves each of these subspaces because it sends each basis vector e_i into the span of the e_0, …, e_{i-1}. You’re absolutely right that upper triangularity is basis-dependent and so is somewhat ‘weird’/‘evil’ (in fact, not even well-defined) as a purported property of maps rather than matrices. What I meant to say was ‘triangularisable’ by analogy with ‘diagonalisable’ — matrices which represent such a map in some basis. Given a linear operator, its matrix representation is triangular when expressed in a basis B if and only if it preserves the standard flag in basis B. * https://en.wikipedia.org/wiki/Flag_(linear_algebra) https://en.wikipedia.org/wiki/Flag_(linear_algebra)
    - Tainnor 2y ago
      > I now feel not only that I don’t like determinants but that I’m allergic even to matrices Determinants are totally basis-invariant, however, similar matrices have the same determinant.

Syzygies 2y ago

The title is a jab at and a response to "Linear Algebra Done Right" (https://linear.axler.net https://linear.axler.net) by Sheldon Axler. Sheldon's book is more suitable for a second course in linear algebra. It's an excellent book, that my more advanced students have enjoyed. Sheldon, however, approaches the subject with some rigid dogma: Determinants are evil. His view is they're not practical to compute at scale. It can be good to learn theory with one's determinant hand tied behind one's back, just as it can be good to learn "constructive" mathematics, but this should not be one's only approach. I used to attend an academic sponsor's day at the math institute MSRI (now SLMath). Afternoons reached to keep us busy. One year Sheldon's wife demonstrated some academic teaching software, and my group was tasked to solve a linear algebra problem that Sheldon had provided. Unaware that everyone was watching us on the big screen, I solved it instantly, using determinants. I still don't understand Sheldon's longer solution.

munchler 2y ago
This is very informative. Thank you for the explanation.
btilly 2y ago
His view is not just that they are not practical to compute at scale. (They actually are practical to compute. Just not with the formula. Use row reduction instead.) His view is that they are presented as an arbitrary piece of magic that people really don't understand. And, exactly because they don't really understand the determinant, they wind up not properly understanding fundamental concepts in linear algebra. Such as Jordan normal form. If your goal is actually to properly understand linear algebra, he's absolutely right. If your goal is to use linear algebra for a wide variety of practical purposes, though, determinants are often essential.
- xanderlewis 2y ago
  Since a significant amount of my knowledge/intuition about linear algebra comes from his book, I now feel not only that I don’t like determinants but that I’m allergic even to matrices. Everything seems so much nicer and friendlier if you think of linear maps between finite-dim vector spaces instead. When I hear, for example, ‘upper triangular matrix’ I have to translate it in my head into something like ‘matrix representing a linear operator that preserves the standard flag’ in order to actually feel like I understand it. Of course, I’m not a programmer or an engineer of any kind, so I have the luxury of not needing the computational efficiency.
  - staunton 2y ago
    What's a "standard flag"? Couldn't find anything about it... I would think "upper triangular" would be a weird/bad notion to talk about in a basis-independent setting (because it depends on the basis).
    - xanderlewis 2y ago
      A ‘flag’ is just a funny name for an increasing (and therefore increasing in dimension) sequence of subspaces. The ‘standard flag’ is the sequence spanned by the standard basis vectors: start with {0}, then things of the form (x,0,…), then (x,y,0,…), and so on. * What I’m saying is that an upper triangular matrix preserves each of these subspaces because it sends each basis vector e_i into the span of the e_0, …, e_{i-1}. You’re absolutely right that upper triangularity is basis-dependent and so is somewhat ‘weird’/‘evil’ (in fact, not even well-defined) as a purported property of maps rather than matrices. What I meant to say was ‘triangularisable’ by analogy with ‘diagonalisable’ — matrices which represent such a map in some basis. Given a linear operator, its matrix representation is triangular when expressed in a basis B if and only if it preserves the standard flag in basis B. * https://en.wikipedia.org/wiki/Flag_(linear_algebra) https://en.wikipedia.org/wiki/Flag_(linear_algebra)
  - Tainnor 2y ago
    > I now feel not only that I don’t like determinants but that I’m allergic even to matrices Determinants are totally basis-invariant, however, similar matrices have the same determinant.