Gilbert Strang taught me how to sanely multiply matrices. His Introduction to Linear Algebra is very approachable. It's wildly different experience compared to linear algebra courses I had on university, it actually makes sense and is fun!
A 2020 Vision of Linear Algebra
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Re: A 2020 Vision of Linear Algebra
#42Have 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 transforma…
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.
Re: A 2020 Vision of Linear Algebra
#43Earlier quoted context omitted.
Is there a better way to incentivize educational institutions to offer free content?
A modest proposal: ban donating to them so they need to radically increase their student body (online or offline) to earn their keep with tuition, instead of relying on the donations of the extremely rich parents of legacy admissions students.
What problem do you think drastically increasing the student body will solve?
Re: A 2020 Vision of Linear Algebra
#44After 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.
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 rememb…
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
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.
Re: A 2020 Vision of Linear Algebra
#45Have 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 transforma…
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.
Re: A 2020 Vision of Linear Algebra
#46Earlier quoted context omitted.
3Blue1Brown's linear algebra animations were fun to watch but they did almost nothing for me except the basic fact that the "linear" part means lines. The rest was effectively preaching to the choir so those that already know linear algebra nodded their heads and idiots like me were still flummoxed
That's interesting. I do think 3B1B's goal is probably to build better intuitions in people who already know it.
Yes, those lectures are good enough for intuitions only. For practice one has to read books. I just hope the viewer knows that. Which I have seen is absent among some people.
Re: A 2020 Vision of Linear Algebra
#47What would you recommend as a good resource for learning about Linear Algebra in 2020?
I am aware of his course on OCW, but wondering is there something more interactive and/or newer than those lectures that has similar quality.
Re: A 2020 Vision of Linear Algebra
#48Another good Linear Algebra book is "Linear Algebra Done Right", which Springer is giving for free right now. Link: https://link.springer.com/book/10.1007/978-3-319-11080-6
Came here to say that. It is a wonderful book, and I think it provides a more "modern" approach than the one presented in the videos.
Re: A 2020 Vision of Linear Algebra
#49What would you recommend as a good resource for learning about Linear Algebra in 2020? I am aware of his course on OCW, but wondering is there something more interactive and/or newer than those lectures that has similar quality.
Frankly, couple with this book, it does hardly get better. You still have 3blue1brown[1] series of video, but it just brush off the surface.
[1] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...
Re: A 2020 Vision of Linear Algebra
#50Another good Linear Algebra book is "Linear Algebra Done Right", which Springer is giving for free right now. Link: https://link.springer.com/book/10.1007/978-3-319-11080-6
There's also a free book "Linear Algebra Done Wrong," which might be worth checking out.