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Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introdu

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transformation in a particular basis is probably the most important and first thing you should learn about matrices ...

dandanua 6y ago
I don't think it was this way in 1820, but I agree with your point. Even though I also use Linear Algebra mostly computationally today, the origin of it is in the geometry and I think this connection should come first. Also, "number crunching" is a boring way to learn things. Though, "matrix way" can be good for engineers.
jacobolus 6y ago
Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years often look down on numerical analysis, statistics, engineering, and natural science, and adopt a position that education of students should be optimized in the direction of helping them prove the maximally general results using the most abstract and technical machinery, with an unfortunate emphasis on symbol twiddling vs. examining concrete examples. By contrast, in the 19th century there was much more of a unified vision and more respect for computations and real-world problems. Gauss himself was employed throughout his career as an astronomer / geodesist, rather than as a mathematician, and arguably his most important work was inventing the method of least squares, which he used for interpreting astronomical observations. With the rise of electronic computers, it is possible that the dominant 2050 vision of linear algebra and the dominant 1900 vision of linear algebra will be closer to each-other than either one is to a 1950 vision from a graduate course in a pure math department.
- auggierose 6y ago
  I believe in applications and theory going hand in hand together and benefiting each other. The computer is an incredibly powerful tool perfectly suited for this purpose. If we resist the urge to just see it as a push-button technology. Viewing matrices as a box of numbers instead of as a representation of a linear transformation leans too much in the direction of push-button for my taste.
  - jacobolus 6y ago
    Gil Strang does not view matrices as “just boxes of numbers”, nor does he teach that view. YMMV, but I find pure mathematicians treat computers as “push-button technology” much more than applied mathematicians.
    - auggierose 6y ago
      I am not disagreeing with you there when it comes to pure mathematicians :-D Edit: But there are of course big exceptions there as well, for example Thomas Hales.
- atrettel 6y ago
  > Pure mathematics departments for at least the past maybe 80 years often look down on numerical analysis, statistics, engineering, and natural science, and adopt a position that education of students should be optimized in the direction of helping them prove the maximally general results using the most abstract and technical machinery, with an unfortunate emphasis on symbol twiddling vs. examining concrete examples. I had this view when I took linear algebra as an undergraduate, but I have gradually changed on the subject over time. I took a standard "linear algebra for scientists and engineers" course but I found it too abstract at the time. The instructor rarely concentrated on examples and applications despite the more applied focus in the course title. Later I came to appreciate the abstraction, since it helped me understand more advanced mathematical topics unrelated to the "number-crunching" I originally associated the topic with. I now think the instructor had a more "unified" approach, but I didn't realize it at the time.
- 6y ago
JoeCamel 6y ago
I also think that this point should be emphasized more. It helped me a lot when I realized it. I also liked the abstract approach to vector spaces. Of course, a matrix is just one way to represent linear transformations and it can also represent other things like a system of linear equations.
tchaffee 6y ago
> Have not watched the videos yet, Then please do. I took several online linear algebra courses from sources I trust and they were pretty bad. Or let's put it another way: I'm a pretty clever guy and I was still left confused. Strang is excellent in the classroom, and I almost even like videos for learning now thanks to him (x1.5 speed is your friend). His videos should not be your only learning source, but judging his course only by the book might result in a lot of learners skipping what I found to be the best course by far. If you want to learn linear algebra, give Strang a try first and you might save a lot of time.
- deleted 6y ago
  [deleted]
gowld 6y ago
This is silly. LA is incredibly rich, broad, and feel area of study. You can't just grab one part of it and say it's the most important and first thing. And it's silly to say that whatever is most important should be first -- the central ideas depend on prepaeatt.
- bo1024 6y ago
  I’m not sure I agree when that one thing is the word linear (half the name). It just feels wrong not to ground the subject in the idea of a linear transformation.
plandis 6y ago
When I was self studying linear algebra I found Strange’s book to be inadequate on its own, you really need the lectures to get the most value out of it. I found going through Linear Algebra Done Right to provide a good counterbalance to Strang’s book+lectures.
Koshkin 6y ago
> a matrix is just the representation of a linear transformation While this view certainly helps intuition at initial stages of learning, it is not "just" that, and computational methods involving matrices are of much more practical importance (similar to being able to add and multiply numbers which we are taught early in life) which is probably why the stress is on them first and foremost. Someone said, "learn to calculate, understanding will come later."

dandanua 6y ago

I don't think it was this way in 1820, but I agree with your point. Even though I also use Linear Algebra mostly computationally today, the origin of it is in the geometry and I think this connection should come first. Also, "number crunching" is a boring way to learn things. Though, "matrix way" can be good for engineers.

jacobolus 6y ago

Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years often look down on numerical analysis, statistics, engineering, and natural science, and adopt a position that education of students should be optimized in the direction of helping them prove the maximally general results using the most abstract and technical machinery, with an unfortunate emphasis on symbol twiddling vs. examining concrete examples. By contrast, in the 19th century there was much more of a unified vision and more respect for computations and real-world problems. Gauss himself was employed throughout his career as an astronomer / geodesist, rather than as a mathematician, and arguably his most important work was inventing the method of least squares, which he used for interpreting astronomical observations. With the rise of electronic computers, it is possible that the dominant 2050 vision of linear algebra and the dominant 1900 vision of linear algebra will be closer to each-other than either one is to a 1950 vision from a graduate course in a pure math department.

auggierose 6y ago
I believe in applications and theory going hand in hand together and benefiting each other. The computer is an incredibly powerful tool perfectly suited for this purpose. If we resist the urge to just see it as a push-button technology. Viewing matrices as a box of numbers instead of as a representation of a linear transformation leans too much in the direction of push-button for my taste.
- jacobolus 6y ago
  Gil Strang does not view matrices as “just boxes of numbers”, nor does he teach that view. YMMV, but I find pure mathematicians treat computers as “push-button technology” much more than applied mathematicians.
  - auggierose 6y ago
    I am not disagreeing with you there when it comes to pure mathematicians :-D Edit: But there are of course big exceptions there as well, for example Thomas Hales.
atrettel 6y ago
> Pure mathematics departments for at least the past maybe 80 years often look down on numerical analysis, statistics, engineering, and natural science, and adopt a position that education of students should be optimized in the direction of helping them prove the maximally general results using the most abstract and technical machinery, with an unfortunate emphasis on symbol twiddling vs. examining concrete examples. I had this view when I took linear algebra as an undergraduate, but I have gradually changed on the subject over time. I took a standard "linear algebra for scientists and engineers" course but I found it too abstract at the time. The instructor rarely concentrated on examples and applications despite the more applied focus in the course title. Later I came to appreciate the abstraction, since it helped me understand more advanced mathematical topics unrelated to the "number-crunching" I originally associated the topic with. I now think the instructor had a more "unified" approach, but I didn't realize it at the time.
6y ago

JoeCamel 6y ago

I also think that this point should be emphasized more. It helped me a lot when I realized it. I also liked the abstract approach to vector spaces. Of course, a matrix is just one way to represent linear transformations and it can also represent other things like a system of linear equations.

tchaffee 6y ago

> Have not watched the videos yet, Then please do. I took several online linear algebra courses from sources I trust and they were pretty bad. Or let's put it another way: I'm a pretty clever guy and I was still left confused. Strang is excellent in the classroom, and I almost even like videos for learning now thanks to him (x1.5 speed is your friend). His videos should not be your only learning source, but judging his course only by the book might result in a lot of learners skipping what I found to be the best course by far. If you want to learn linear algebra, give Strang a try first and you might save a lot of time.

deleted 6y ago
[deleted]

gowld 6y ago

This is silly. LA is incredibly rich, broad, and feel area of study. You can't just grab one part of it and say it's the most important and first thing. And it's silly to say that whatever is most important should be first -- the central ideas depend on prepaeatt.

bo1024 6y ago
I’m not sure I agree when that one thing is the word linear (half the name). It just feels wrong not to ground the subject in the idea of a linear transformation.

plandis 6y ago

When I was self studying linear algebra I found Strange’s book to be inadequate on its own, you really need the lectures to get the most value out of it. I found going through Linear Algebra Done Right to provide a good counterbalance to Strang’s book+lectures.

Koshkin 6y ago

> a matrix is just the representation of a linear transformation While this view certainly helps intuition at initial stages of learning, it is not "just" that, and computational methods involving matrices are of much more practical importance (similar to being able to add and multiply numbers which we are taught early in life) which is probably why the stress is on them first and foremost. Someone said, "learn to calculate, understanding will come later."