4 ms·

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 a

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.
- xanderlewis 2y ago
  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).

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.

xanderlewis 2y ago
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).