6 ms·

In college I was taught Linear Algebra from the operator point of view, rather than with matrices. That way theorems are clearer and the student's understanding

In college I was taught Linear Algebra from the operator point of view, rather than with matrices. That way theorems are clearer and the student's understanding is deeper, but for applications it's better to study from the matrix point of view and with lots of examples. Kuttler's book was refreshing in that sense. His other books are excellent, too. If you have been studying pure math (or french-style applied math which is just pure math with a concentration in Analysis) they are a light and fun complementary read.

mhh__ 6y ago
Although I understand that matrix-soup is kind of the entropic endstate of all high school mathematics pedagogy, I think it is a real tragedy: In the UK especially even the best and brightest barely touch any real mathematics until after they leave secondary school so they leave with almost literally no idea of what university mathematics consists of. It's all well and good training people to be engineers, but the universities end up teaching the whole syllabus to them again in about 6 weeks. It's just shit. The only reason why I am now doing theoretical physics (I was in the dumb group initially and worked my way up largely by myself) is because I read a calculus textbook by accident and got hooked when I was 14. Even when I made it to the top of the pile I still wasn't allowed to do anything more than calculus because the module system means we had to choose as a class whether to do group theory or not.
- 2wrist 6y ago
  Good luck to you. I wish you every success. The education in this country has been shit for such a long time. It is and has been pretty devastating for so long. Apologies for the negative waves.
  - mhh__ 6y ago
    No negativity measured, actually
    - deleted 6y ago
      [deleted]
- cpp_frog 6y ago
  I understand, I too had a similar experience in high school. My comment was largely referring to my context: This semester I'll finish a mathematical engineering degree (in my country engs. are 12 semesters long) and the abstract way is rewarding but without applications can only be endured so far. It does not help that a large fraction of mathematical engineers after they graduate end up doing some software work consisting in statistics, optimization or some other advanced mathetical concept. With a dropout rate of ~80%, not many people are willing to put themselves through rigorous analysis classes on top of the engineering requirements for so many years to get a job doing something you could have learned in a more practical way. But we are very well paid and the unemployment rate is 0% since there are maybe 10 graduates in the whole country each year.
lr1970 6y ago
The best, by far, book on Linear Algebra that elegantly teaches it from Vector Spaces and Linear Operators point of view is Paul Halmos "Finite-Dimensional Vector Spaces" For instance, the way Halmos introduces the determinant of a matrix (or an operator) is the most consistent, elegant and simple way I ever encountered. OTOH, in Kenneth Kuttler's LinAlg books the determinant is pulled out of the thin air like in 1000+ other similar books.
- cpp_frog 6y ago
  Thanks, I'll look it up. The best textbook from which I studied (operators) was Elon Lima's Algebra Linear. Sadly the only physical copies are sold in Brazil.
  - spekcular 6y ago
    If you want to see the matrix point of view done well, there's Linear Algebra Done Wrong: https://www.math.brown.edu/streil/papers/LADW/LADW.html https://www.math.brown.edu/streil/papers/LADW/LADW.html. You can read a bit about the motivation for doing it that way on that website. The title is a reference to a somewhat well-known book, Linear Algebra Done Right, which avoids using determinants to develop the theory (resulting in a somewhat novel/cleaner presentation). It's unfortunately not freely available online (published by Springer – I would suspect most university students can get it freely through their library's website, however).
    - Tomte 6y ago
      LADR was freely available at Springer at least at some point in time. Under their open access program. I couldn't find it again in a few minutes' search, so it may be gone.
  - in9 6y ago
    Ah yes, a fellow Brazillian. I've alwys found Elon's book on Linear Algebra a masterpiece. Coupled with the exercises book, going through it is an eye opening experience.
- 6y ago
fangyrn 6y ago
Could you please talk about alternative ways of learning about linear algebra?

mhh__ 6y ago

Although I understand that matrix-soup is kind of the entropic endstate of all high school mathematics pedagogy, I think it is a real tragedy: In the UK especially even the best and brightest barely touch any real mathematics until after they leave secondary school so they leave with almost literally no idea of what university mathematics consists of. It's all well and good training people to be engineers, but the universities end up teaching the whole syllabus to them again in about 6 weeks. It's just shit. The only reason why I am now doing theoretical physics (I was in the dumb group initially and worked my way up largely by myself) is because I read a calculus textbook by accident and got hooked when I was 14. Even when I made it to the top of the pile I still wasn't allowed to do anything more than calculus because the module system means we had to choose as a class whether to do group theory or not.

2wrist 6y ago
Good luck to you. I wish you every success. The education in this country has been shit for such a long time. It is and has been pretty devastating for so long. Apologies for the negative waves.
- mhh__ 6y ago
  No negativity measured, actually
  - deleted 6y ago
    [deleted]
cpp_frog 6y ago
I understand, I too had a similar experience in high school. My comment was largely referring to my context: This semester I'll finish a mathematical engineering degree (in my country engs. are 12 semesters long) and the abstract way is rewarding but without applications can only be endured so far. It does not help that a large fraction of mathematical engineers after they graduate end up doing some software work consisting in statistics, optimization or some other advanced mathetical concept. With a dropout rate of ~80%, not many people are willing to put themselves through rigorous analysis classes on top of the engineering requirements for so many years to get a job doing something you could have learned in a more practical way. But we are very well paid and the unemployment rate is 0% since there are maybe 10 graduates in the whole country each year.

lr1970 6y ago

The best, by far, book on Linear Algebra that elegantly teaches it from Vector Spaces and Linear Operators point of view is Paul Halmos "Finite-Dimensional Vector Spaces" For instance, the way Halmos introduces the determinant of a matrix (or an operator) is the most consistent, elegant and simple way I ever encountered. OTOH, in Kenneth Kuttler's LinAlg books the determinant is pulled out of the thin air like in 1000+ other similar books.

cpp_frog 6y ago
Thanks, I'll look it up. The best textbook from which I studied (operators) was Elon Lima's Algebra Linear. Sadly the only physical copies are sold in Brazil.
- spekcular 6y ago
  If you want to see the matrix point of view done well, there's Linear Algebra Done Wrong: https://www.math.brown.edu/streil/papers/LADW/LADW.html https://www.math.brown.edu/streil/papers/LADW/LADW.html. You can read a bit about the motivation for doing it that way on that website. The title is a reference to a somewhat well-known book, Linear Algebra Done Right, which avoids using determinants to develop the theory (resulting in a somewhat novel/cleaner presentation). It's unfortunately not freely available online (published by Springer – I would suspect most university students can get it freely through their library's website, however).
  - Tomte 6y ago
    LADR was freely available at Springer at least at some point in time. Under their open access program. I couldn't find it again in a few minutes' search, so it may be gone.
- in9 6y ago
  Ah yes, a fellow Brazillian. I've alwys found Elon's book on Linear Algebra a masterpiece. Coupled with the exercises book, going through it is an eye opening experience.
6y ago

fangyrn 6y ago

Could you please talk about alternative ways of learning about linear algebra?