3 ms·

Regardless of the usefulness of category theory or not, the author seems to imply that "traditional linear algebra" is about matrices. This is not true. This m

Regardless of the usefulness of category theory or not, the author seems to imply that "traditional linear algebra" is about matrices. This is not true. This may be true from an engineering perspective, or maybe if you've studied in the US, but over here, linear algebra starts with fields, vector spaces, homomorphisms. We establish pretty early on that matrices and linear transformations are in essence the same thing (up to choice of basis), and then we go on to prefer coordinate-free proofs. But, of course, we do discuss the computational aspect (i.e. the Gauss algorithm) because this is exactly what makes linear algebra so useful: many questions can be answered extremely efficiently. Whether you want to frame things in the language of category theory or not, is up to anyone, but the "standard mathematical treatment" is quite beautiful and intriguing already (if I want to feel inspired, I open up my copy of Halmos's "Finite Dimensional Vector Spaces", not necessarily the best introduction for someone who is new, but very beautifully written for someone who has already seen linear algebra).

buddhiajuke 6y ago
You’re not wrong. OTOH the other day I heard someone in the Netflix doc about Alberto Nisman’s murder that described a terrorist network as a “matrix”. I grumped as card-carrying pedants are duty-bound to, but then realized that if there’s a 1:1 correspondence between graphs and (adjacency, incidence) matrices, then there’s very little loss of meaning in referring to graphs as “matrices”. Maybe in some communication contexts it’s even clearer.
- uglycoyote 6y ago
  That's funny, I would have just chalked it up to "they are using a different sense of the word matrix", knowing that most dictionaries contain multiple definitions of the same word. But nice work making this mental leap, I have never thought about it! But it's true, you can envision it as a NxN matrix of N terrorists, where the number in cell i,j represent's the strength of the relationship (it's a weighted graph!), and that might actually be useful for something. I don't doubt that "matrix" might actually exist in numeric form in some CIA or military computer somewhere.
  - a9h74j 6y ago
    A few months ago I was imagining having a square matrix representation for fractions/attenuation of viral loads (e.g. per week) being transferred among a large population. I pictured a sum of isolated groups, after a sort, causing this larger matrix to be composed of a set of on-diagonal square submatrices.
    - uglycoyote 6y ago
      Yes I would think that you are dealing with sparse matrices with little clusters of nonzero values in a model like this. One could say that I'm "part of a terrorist matrix", and that my value in the matrix is zero. We are all part of a terrorist matrix. I hope the CIA is not reading this, because they probably are not mathy enough to know what I'm talking about, and they may just take my quoted statement at face value, but in a mathematical sense we probably all actually have some nonzero value in our rows and columns of the terrorist matrix. You know someone, who knows someone, who knows someone, who is a terrorist. It really just depends how many people and how many degrees of separation you want to include in your matrix.
  - microcolonel 6y ago
    Maybe the term Disposition Matrix was more precise and less euphemistic than we were concerned it was. I guess you could compute a kill list from a genuine matrix of dispositions. https://en.wikipedia.org/wiki/Disposition_Matrix https://en.wikipedia.org/wiki/Disposition_Matrix
nerdponx 6y ago
This is not true. This may be true from an engineering perspective, or maybe if you've studied in the US, but over here, linear algebra starts with fields, vector spaces, homomorphisms. We establish pretty early on that matrices and linear transformations are in essence the same thing (up to choice of basis), and then we go on to prefer coordinate-free proofs. But, of course, we do discuss the computational aspect (i.e. the Gauss algorithm) because this is exactly what makes linear algebra so useful: many questions can be answered extremely efficiently. This is how I was taught linear algebra in the USA too. Almost to a fault, where I lacked practical intuition for how matrices and vectors of real numbers work and had to re-learn that stuff later.
wenc 6y ago
Just curious, what country would this be? I know many European countries favor a more theoretical approach. I definitely learned linear algebra from the numerical and matrix perspective (though yes, we did cover the idea of vector spaces and linear transformations too). In engineering school, the focus tended to be on the craft rather than the theory, and in retrospect I think it was the right approach for engineering majors, whose primary concern was execution. It helped engineers use MATLAB to solve problems. The syllabus looked like this: https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/syllabus/ https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb... And this: https://math.nyu.edu/media/math/filer_public/a4/00/a4008ffa-f5b0-48df-9b76-357d32397009/math-ua_140_fall_2019.pdf https://math.nyu.edu/media/math/filer_public/a4/00/a4008ffa-... A syllabus for math majors might look quite different. The US approach to linear algebra continued to be useful for me in grad school as an engineering major (I have an engineering Ph.D.). To be honest, for what I was/am doing, I don't think I would have derived much benefit from a more theoretical approach (but that's just me. People who work closer to theory may have a different opinion). In my work, math has a retrospective role -- I first build something and later recognize patterns that fit a certain theory. For theory-builders however, math has a constructive role. Theory builders however are relatively rare.
- Tainnor 6y ago
  Germany, but I think it's similar across much of Europe. The US has a different university structure: the way I understand it, many people don't necessarily choose a major early on or can easily switch it which is why engineers, mathematicians, biologists, etc. often take the same course. Please correct me if I'm wrong. Over here, this is much rarer. Sometimes computer scientists or physicists take some of the same courses as mathematicians, but hardly engineers. Therefore, we can probably afford to be more theoretical right away (although the computer scientists sometimes complain xD). I can totally understand that for someone with a focus on application theory is less important, and that some people can get more excited about what you can do with matrices e.g. in computer graphics or machine learning than about abstract morphisms and rigorous proofs. Incidentally, over here I sometimes feel we learn stuff the other way around: first we hear all the theory, but it's only in the later ODE and numerical methods courses that we actually get substantial practice in calculating more complicated integrals or learn about e.g. QR factorisation or Simpson's rule (although we do prove error bounds by then).

buddhiajuke 6y ago

You’re not wrong. OTOH the other day I heard someone in the Netflix doc about Alberto Nisman’s murder that described a terrorist network as a “matrix”. I grumped as card-carrying pedants are duty-bound to, but then realized that if there’s a 1:1 correspondence between graphs and (adjacency, incidence) matrices, then there’s very little loss of meaning in referring to graphs as “matrices”. Maybe in some communication contexts it’s even clearer.

uglycoyote 6y ago
That's funny, I would have just chalked it up to "they are using a different sense of the word matrix", knowing that most dictionaries contain multiple definitions of the same word. But nice work making this mental leap, I have never thought about it! But it's true, you can envision it as a NxN matrix of N terrorists, where the number in cell i,j represent's the strength of the relationship (it's a weighted graph!), and that might actually be useful for something. I don't doubt that "matrix" might actually exist in numeric form in some CIA or military computer somewhere.
- a9h74j 6y ago
  A few months ago I was imagining having a square matrix representation for fractions/attenuation of viral loads (e.g. per week) being transferred among a large population. I pictured a sum of isolated groups, after a sort, causing this larger matrix to be composed of a set of on-diagonal square submatrices.
  - uglycoyote 6y ago
    Yes I would think that you are dealing with sparse matrices with little clusters of nonzero values in a model like this. One could say that I'm "part of a terrorist matrix", and that my value in the matrix is zero. We are all part of a terrorist matrix. I hope the CIA is not reading this, because they probably are not mathy enough to know what I'm talking about, and they may just take my quoted statement at face value, but in a mathematical sense we probably all actually have some nonzero value in our rows and columns of the terrorist matrix. You know someone, who knows someone, who knows someone, who is a terrorist. It really just depends how many people and how many degrees of separation you want to include in your matrix.
- microcolonel 6y ago
  Maybe the term Disposition Matrix was more precise and less euphemistic than we were concerned it was. I guess you could compute a kill list from a genuine matrix of dispositions. https://en.wikipedia.org/wiki/Disposition_Matrix https://en.wikipedia.org/wiki/Disposition_Matrix

nerdponx 6y ago

This is not true. This may be true from an engineering perspective, or maybe if you've studied in the US, but over here, linear algebra starts with fields, vector spaces, homomorphisms. We establish pretty early on that matrices and linear transformations are in essence the same thing (up to choice of basis), and then we go on to prefer coordinate-free proofs. But, of course, we do discuss the computational aspect (i.e. the Gauss algorithm) because this is exactly what makes linear algebra so useful: many questions can be answered extremely efficiently. This is how I was taught linear algebra in the USA too. Almost to a fault, where I lacked practical intuition for how matrices and vectors of real numbers work and had to re-learn that stuff later.

wenc 6y ago

Just curious, what country would this be? I know many European countries favor a more theoretical approach. I definitely learned linear algebra from the numerical and matrix perspective (though yes, we did cover the idea of vector spaces and linear transformations too). In engineering school, the focus tended to be on the craft rather than the theory, and in retrospect I think it was the right approach for engineering majors, whose primary concern was execution. It helped engineers use MATLAB to solve problems. The syllabus looked like this: https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/syllabus/ https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb... And this: https://math.nyu.edu/media/math/filer_public/a4/00/a4008ffa-f5b0-48df-9b76-357d32397009/math-ua_140_fall_2019.pdf https://math.nyu.edu/media/math/filer_public/a4/00/a4008ffa-... A syllabus for math majors might look quite different. The US approach to linear algebra continued to be useful for me in grad school as an engineering major (I have an engineering Ph.D.). To be honest, for what I was/am doing, I don't think I would have derived much benefit from a more theoretical approach (but that's just me. People who work closer to theory may have a different opinion). In my work, math has a retrospective role -- I first build something and later recognize patterns that fit a certain theory. For theory-builders however, math has a constructive role. Theory builders however are relatively rare.

Tainnor 6y ago
Germany, but I think it's similar across much of Europe. The US has a different university structure: the way I understand it, many people don't necessarily choose a major early on or can easily switch it which is why engineers, mathematicians, biologists, etc. often take the same course. Please correct me if I'm wrong. Over here, this is much rarer. Sometimes computer scientists or physicists take some of the same courses as mathematicians, but hardly engineers. Therefore, we can probably afford to be more theoretical right away (although the computer scientists sometimes complain xD). I can totally understand that for someone with a focus on application theory is less important, and that some people can get more excited about what you can do with matrices e.g. in computer graphics or machine learning than about abstract morphisms and rigorous proofs. Incidentally, over here I sometimes feel we learn stuff the other way around: first we hear all the theory, but it's only in the later ODE and numerical methods courses that we actually get substantial practice in calculating more complicated integrals or learn about e.g. QR factorisation or Simpson's rule (although we do prove error bounds by then).