Comment by Syzygies | Hacker News Reader

Comment by Syzygies | Hacker News Reader

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The title is a jab at and a response to "Linear Algebra Done Right" (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.

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.

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.

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).

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)

> 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.

Yeah, and I think that’s enough motivation for determinants being important beyond as a computational trick. I was being somewhat hyperbolic.

Seeing the determinant as the unique solution to the problem of assigning a scalar to each linear map/matrix in such a way that several natural axioms are satisfied is a very nice way to motivate it. Along with the fact that it, along with the trace, can be calculated from the eigenvectors (from which the basis-invariance is clear).

> Use row reduction instead

Or, if you're using exact arithmetic (e.g. cryptography), you should use something like the Bareiss algorithm - Gaussian elimination is only polynomial if you use floats: https://en.m.wikipedia.org/wiki/Bareiss_algorithm

The Leibniz formula is totally impractical for calculating determinants, but it is sometimes useful in proofs due to its symmetric form.

I guess if you just throw a definition of the determinant at people and leave it at that, I can understand that it doesn't make sense. But that's not how I was introduced to the concept (I learned about abstract vector spaces and transformations and how they are represented by matrices - and only then about determinants and their properties and different ways of computing them). That's why Axler's critique rings somewhat hollow to me - maybe it's different if you learn about linear algebra mostly in an engineering context as opposed to pure maths.

For theoretical purposes, determinants are important because they constitute a homomorphism between the general linear group and the multiplicative group of the underlying field, and the kernel of this homomorphism is the special linear group. This leads to some very short and elegant proofs.

Was V.I. Arnold wrong when he said everything a student needs to know about determinants is derived from geometric considerations? It seems true to me that, outside of advanced study linear or abstract algebra, this is the one perspective which can at least dispel the magic.

Yes, in real space the determinant is the signed volume of the image of the unit cube.

Volume is base x height. Height is a linear function in the vector that rises above the base; the usual properties of the determinant follow from this.

This is very informative. Thank you for the explanation.

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