[auggierose]

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 ...

[jacobolus]

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.

[vlasev]

I believe this is largely because in the field of mathematics, Linear algebra is just the seed that sprouts the growth of other very very useful mathematical subjects like Abstract Algebra, Functional Analysis and so on. Linear Algebra is used as a stepping stone to more general theories that are also super useful.

Take Hilbert spaces for example. They are based on linear algebra. They are quite general and you might argue that there's a lot of symbol twiddling there. However, Hilbert spaces are/were essential in the study of Quantum Mechanics, which we can argue is a very important topic.

And if you only stick with matrices and numerics, you're bound to get stuck in the numbers and details and miss the big picture. A lot of results are much cleaner to obtain once you divorce yourself from the concrete world of matrix representation.

Of course, we should probably have the best of both worlds. I'm not saying applications are unimportant. Take something like signal processing, which relies heavily on both numerics and general theory.

So I'd like to add something to your point. Math departments optimize the education of math students towards the more general, and perhaps students not interested in pursuing pure math should have course-work that reflects that.

[atrettel]

> 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.

[auggierose]

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]

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]

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.

[BeetleB]

If you want to get into serious physics/engineering, the abstract aspect of linear algebra is much more important than the boring computational mechanics. And quite a bit of those computational mechanics lead you astray when you go to infinite dimensions.

[slavik81]

How might they lead you astray? It almost sounds like you have an example in mind.

[mattkrause]

Indeed, Strang’s textbook starts with “I believe the teaching of linear algebra has become too abstract.”

He mentions this sentiment in a lot of interviews and things too.

[tchaffee]

> 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.

[Koshkin]

> 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]

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.

[JoeCamel]

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.

[gowld]

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]

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]

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.