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After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of

by brmgb 6y ago

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

chubot 6y ago
Yeah I would call this the engineering approach (matrices) vs the mathematical approach (algebra). I took like 3-4 courses in the US involving the engineering approach, starting in high school and continuing through the college as a CS major. That was all that was required. But I also like algebra, so I happened to take a 400-level course that only math majors take my senior of college. And then I got the group theory / vector space view on it. I don't think 95% of CS majors got that. I don't think one is better than the other, but they should have tried to balance it out more. It helps to understand both viewpoints. (If you haven't seen the latter, then picture a 300-page text on linear algebra that doesn't mention matrices at all. It's all linear transformations and spaces.) What country were you taught in? Wild guess: France?
ojnabieoot 6y ago
I would not describe his approach as “the US approach” but it is a pretty standard approach to introducing linear algebra to engineers, which is the theme of the course. I was also taught linear algebra this way, by an applied mathematician with a background in chemical engineering: - start by solving Ax=b with row reduction - develop theorems about linear independence and spanning sets of vectors based on these exercises - introduce the determinant from the perspective of linear systems (rather than eg geometry or group theory) - eigenvectors and eigenvalues Later I switched from physics to math and TAed a more “algebraic” approach involving groups/rings/fields. But the matrix-first approach was more helpful for both my physics coursework and later courses in numerical linear algebra.
tadhgds 6y ago
I can't say whether or not it is the standard approach but I do know that it is very common in many countries to teach a linear algebra course that is heavy on matrix operations, that you can come away believing that linear algebra is somehow _about_ matrices and their operations. I know many in my university class seemed to believe that. A book I enjoyed is Axler's Linear Algebra Done Right[0], in which, if I remember correctly, doesn't contain a single matrix. [0]https://zhangyk8.github.io/teaching/file_spring2018/linear_algebra_done_right.pdf https://zhangyk8.github.io/teaching/file_spring2018/linear_a...
- impendia 6y ago
  Yeah, math professor here, this drove me crazy. For example, I remember looking at the linear algebra book my department had used previously. Early on, it introduced the concept of the transpose of a matrix: https://en.wikipedia.org/wiki/Transpose https://en.wikipedia.org/wiki/Transpose Superficially, it looks like something good to introduce. It is fodder for easy homework exercises, and there is a satisfyingly long list of formal properties satisfied. But why? What does the transpose mean? For what sort of problem would you want to compute it? There are good answers to these questions (see the "transpose of a linear map" section of the Wikipedia article I linked), but they are not easy for a beginner to the subject to appreciate.
- andy_wrote 6y ago
  I've recently started going through Axler carefully and doing the problems, a quarantine activity I guess, and have been enjoying it. I actually learned about this book on an older HN post. It does have plenty of matrices. The main thing it really does is avoid determinants until the very end. The determinant is certainly something I remember learning as a kind of rote operation, without really understanding any intuition behind why you'd multiply and add these numbers in this particular way. I still feel lacking in "feel" here, which is why I suppose I'm going through Axler now.
- threatofrain 6y ago
  IMO Axler's book should be read either during or after you take an introductory course on Linear Algebra. > You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, this encounter will focus on abstract vector spaces and linear maps.
dntbnmpls 6y ago
> After watching this and having read the comment, I am quite puzzled by the approach American seems to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? The US is a very big place. I doubt there is an american approach to linear algebra. We really don't have a single approach to anything. Different schools and majors probably approach the topic differently. My college had a linear algebra course specifically crafted for CS majors and engineers. I took that and it did focus on matrices. It was also the only math class that required programming. I believe math majors had their own linear algebra course. > My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later. Different strokes for different folks. If it worked out for you that's all that matters.
- Myrmornis 6y ago
  > Different strokes for different folks. If it worked out for you that's all that matters. That sort of relativism is trite, specious, and insincere. If you don’t think it matters then why are you participating in the discussion?
  - dntbnmpls 6y ago
    > That sort of relativism is trite, specious, and insincere. It isn't. I don't think you understood what I wrote. What's important is the ends, not the means. As long as the OP got what he needed out of linear algebra, then that's what matters. > If you don’t think it matters then why are you participating in the discussion? What is this "it" you are talking about? Be more specific and precise otherwise you don't add anything to the discussion.
    - Myrmornis 6y ago
      I'm sorry, but I really dislike the "whatever works for you" sentiment. It's antithetical to the spirit of engineering / technical discussion, which are predicated on the idea that there are better and worse ways to do the thing in question. > As long as the OP got what he needed out of linear algebra, then that's what matters. I don't think I even agree with this. I might agree with > As long as the method of teaching linear algebra improves the mean educational outcome in a statistically significant fashion then that's what matters. But that's what Strang's getting at -- better and worse ways of teaching Linear Algebra. It's not the case that any method is OK just because it worked for some European dude on the internet with poorly concealed anti-American feelings (I'm from Europe, I'm allowed to say this :) ).
einpoklum 6y ago
There's nothing particularly linear about groups.
- brmgb 6y ago
  No but the curriculum goes from groups to fields and from fields to vector spaces.
currymj 6y ago
In general math departments in the US are less frightened of accidentally teaching the students something useful.
eximius 6y ago
American here. We started with the group theory and vector space approach, though the group theory was fairly limited to just enough for vector spaces as there was a separate set of algebra classes. It's not universal.

by brmgb 6y ago

chubot 6y ago

Yeah I would call this the engineering approach (matrices) vs the mathematical approach (algebra). I took like 3-4 courses in the US involving the engineering approach, starting in high school and continuing through the college as a CS major. That was all that was required. But I also like algebra, so I happened to take a 400-level course that only math majors take my senior of college. And then I got the group theory / vector space view on it. I don't think 95% of CS majors got that. I don't think one is better than the other, but they should have tried to balance it out more. It helps to understand both viewpoints. (If you haven't seen the latter, then picture a 300-page text on linear algebra that doesn't mention matrices at all. It's all linear transformations and spaces.) What country were you taught in? Wild guess: France?

ojnabieoot 6y ago

I would not describe his approach as “the US approach” but it is a pretty standard approach to introducing linear algebra to engineers, which is the theme of the course. I was also taught linear algebra this way, by an applied mathematician with a background in chemical engineering: - start by solving Ax=b with row reduction - develop theorems about linear independence and spanning sets of vectors based on these exercises - introduce the determinant from the perspective of linear systems (rather than eg geometry or group theory) - eigenvectors and eigenvalues Later I switched from physics to math and TAed a more “algebraic” approach involving groups/rings/fields. But the matrix-first approach was more helpful for both my physics coursework and later courses in numerical linear algebra.

tadhgds 6y ago

I can't say whether or not it is the standard approach but I do know that it is very common in many countries to teach a linear algebra course that is heavy on matrix operations, that you can come away believing that linear algebra is somehow _about_ matrices and their operations. I know many in my university class seemed to believe that. A book I enjoyed is Axler's Linear Algebra Done Right[0], in which, if I remember correctly, doesn't contain a single matrix. [0]https://zhangyk8.github.io/teaching/file_spring2018/linear_algebra_done_right.pdf https://zhangyk8.github.io/teaching/file_spring2018/linear_a...

impendia 6y ago
Yeah, math professor here, this drove me crazy. For example, I remember looking at the linear algebra book my department had used previously. Early on, it introduced the concept of the transpose of a matrix: https://en.wikipedia.org/wiki/Transpose https://en.wikipedia.org/wiki/Transpose Superficially, it looks like something good to introduce. It is fodder for easy homework exercises, and there is a satisfyingly long list of formal properties satisfied. But why? What does the transpose mean? For what sort of problem would you want to compute it? There are good answers to these questions (see the "transpose of a linear map" section of the Wikipedia article I linked), but they are not easy for a beginner to the subject to appreciate.
andy_wrote 6y ago
I've recently started going through Axler carefully and doing the problems, a quarantine activity I guess, and have been enjoying it. I actually learned about this book on an older HN post. It does have plenty of matrices. The main thing it really does is avoid determinants until the very end. The determinant is certainly something I remember learning as a kind of rote operation, without really understanding any intuition behind why you'd multiply and add these numbers in this particular way. I still feel lacking in "feel" here, which is why I suppose I'm going through Axler now.
threatofrain 6y ago
IMO Axler's book should be read either during or after you take an introductory course on Linear Algebra. > You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, this encounter will focus on abstract vector spaces and linear maps.

dntbnmpls 6y ago

> After watching this and having read the comment, I am quite puzzled by the approach American seems to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? The US is a very big place. I doubt there is an american approach to linear algebra. We really don't have a single approach to anything. Different schools and majors probably approach the topic differently. My college had a linear algebra course specifically crafted for CS majors and engineers. I took that and it did focus on matrices. It was also the only math class that required programming. I believe math majors had their own linear algebra course. > My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later. Different strokes for different folks. If it worked out for you that's all that matters.

Myrmornis 6y ago
> Different strokes for different folks. If it worked out for you that's all that matters. That sort of relativism is trite, specious, and insincere. If you don’t think it matters then why are you participating in the discussion?
- dntbnmpls 6y ago
  > That sort of relativism is trite, specious, and insincere. It isn't. I don't think you understood what I wrote. What's important is the ends, not the means. As long as the OP got what he needed out of linear algebra, then that's what matters. > If you don’t think it matters then why are you participating in the discussion? What is this "it" you are talking about? Be more specific and precise otherwise you don't add anything to the discussion.
  - Myrmornis 6y ago
    I'm sorry, but I really dislike the "whatever works for you" sentiment. It's antithetical to the spirit of engineering / technical discussion, which are predicated on the idea that there are better and worse ways to do the thing in question. > As long as the OP got what he needed out of linear algebra, then that's what matters. I don't think I even agree with this. I might agree with > As long as the method of teaching linear algebra improves the mean educational outcome in a statistically significant fashion then that's what matters. But that's what Strang's getting at -- better and worse ways of teaching Linear Algebra. It's not the case that any method is OK just because it worked for some European dude on the internet with poorly concealed anti-American feelings (I'm from Europe, I'm allowed to say this :) ).

einpoklum 6y ago

There's nothing particularly linear about groups.

brmgb 6y ago
No but the curriculum goes from groups to fields and from fields to vector spaces.

currymj 6y ago

In general math departments in the US are less frightened of accidentally teaching the students something useful.

eximius 6y ago

American here. We started with the group theory and vector space approach, though the group theory was fairly limited to just enough for vector spaces as there was a separate set of algebra classes. It's not universal.