Comment by olooney | Hacker News Reader

Comment by olooney | Hacker News Reader

I did Strang's linear algebra course[0] (link goes to Youtube playlist of lectures) several years after graduating and recommend it highly. I was looking for refresher but I gained a deeper understanding of several important concepts; in particular it's fair to say I barely understood, or perhaps even misunderstood, SVD until Strang. If you're not sure if you need something like that, I suggest doing something like testing yourself on this video[1] or the MIT problem sets[2] it's easy to tell yourself that you "know" linear algebra when it would be closer to the truth to say that you used to know linear algebra, but can't answer even basic questions today. After Strang, Golub's book on Matrix Computations is also really incredible.[3]

[0]: https://www.youtube.com/watch?v=ZK3O402wf1c&list=PL49CF3715C...

[1]: https://www.youtube.com/watch?v=Cll03FUxjuk

[2]: https://ocw.mit.edu/courses/mathematics/18-650-statistics-fo...

[3]: https://www.amazon.com/Computations-Hopkins-Studies-Mathemat...

So grateful something mentioned this man. Strang is just amazing. When I first moved to Cambridge, his corresponding book was a relatively expensive prospect, and I was rather serious about 18.06, so ascertaining its book was very important to me. He was gracious enough to gift me a copy that I still have and cherish to this day. Some real moments of thrilling discovery happened for me, it was exhilarating, though trite as they may be! Like in implementing a program for general inversion of any MxN matrix, one would typically perform the Gaussian elimination (going down) and then the Jordan (going back up) and finally divide by the scalars in the pivot columns. But, as it turns out, it's a much simpler program if you do Gauss elimination, literally rotate all matrices by 180 degrees, do Gauss elimination again, then rotate everything back, and then address the non-unit pivot columns. src: https://github.com/wittedhaddock/AlgebraicCircumscriptions/b...

> general inversion of any MxN matrix

??! is there such a thing ? only nonsingular square matrices can be inverted. a general mxn matrix may have a “left inverse” or a “right inverse”, but i don’t think your code is computing that.

There is such a thing as a generalized inverse, in which you can invert non-square or non-full rank matrices and the generalized inverse meets many (but not all) of the properties of an inverse. The tough part is that the agreed upon set of properties does not create a unique solution for a generalized inverse like it does for the inverse, so there are multiple possibilities when someone says generalized inverse. However, the most popular is probably the Moore-Penrose inverse: https://en.wikipedia.org/wiki/Moore–Penrose_inverse

thank you very much for this !!

You're 100% correct, the algo only handles square matrices! Typo. So, disclaimer, where m === n :) thanks!

I like Strang and have worked through some of his books but when I was first learning Linear Algebra, I preferred learning from Dr. Aviv Censor's videos on YouTube[0] from when he was teaching at Technion. It's a few dozen videos where he spends a lot of time going through each component of an introductory course on Linear Algebra.

[0]: https://www.youtube.com/watch?v=aefKXYYXT6I&list=PLW3u28VuDA...

/aside As someone who's also refreshing maths: What's the point of investing in studying hard, if I end up forgetting? Is there some aspect of maths I should focus on, that is an enduring investment?

My feelings is that getting an intuitive understanding is key, so that it becomes part of myself... but I find proofs don't give me this (they show me that it is true, but not why it is that it is true - if that makes sense). Derivations seem important, because then I can re-derive when I forget (though, I must recall the "tricks" of the derivation, and also know the operations used). General skills can persist, if they get ongoing exercise (e.g. methodicalness, care with definitions, close reading, organization to cope with complex and multi-layered problems).

Finally - and awfully - I now think mathematics is like a language, not so much in the sense of communication, but by being full of special cases, exceptions, "abuse of notation". Becoming fluent in a language takes much practice, and may be impossible without special talent. But once achieved, is never fully forgotten, and quickly regained.

The payoff is indeed learning the intuition, but also learning how to prove things and learning how to learn math. The content itself is only as important as whatever field you're in, or whatever application you're studying that requires that specific mathematics. But the "thinking like a mathematician" and knowing how to learn new math is the valuable skill you'll gain.

Another great "intermediate" textbook (in my opinion) is Trefethen's Numerical Linear Algebra [1]. Much more readable than Golub's I would say, which is more like a reference than a textbook.

[1]: https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet...

Strongly agree. This was actually the text book when I took numerical methods at MIT. As far as math texts go, it is definitely a joy to read.

Strang just published a new book called "Linear Algebra and Learning From Data" [1] which I only just started but find to be quite enjoyable so far. It's simultaneously conversational but also quite terse (similar to "All of Statistics"). There are many advanced and very contemporary applications covered in the book, with a focus towards machine learning.

[1]: https://www.amazon.com/Linear-Algebra-Learning-Gilbert-Stran...

I love Strang's conversational style. If you've ever heard him give a lecture, you hear him when reading his textbooks.

Q&A with Strang: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3060226/

Absolutely this!

"Now this is a matrix you wouldn't want to meet in a dark alley!" - Gil

Golub's Matrix Computations is definitely a gold standard, but I wouldn't characterize it as something you read so much as reference :)

When taking Lin Alg in undergrad, I attended the first 3 classes. My friend then showed me the youtube series from Strang; I never went to class again and learned it all.

I partially credit Strang for getting me an A in Linear Algebra. His lectures and the book from David C Lay that I thought was pretty good.

The PSET posted is for Statistics for Applications. I was wondering why it looked so different from when I went through it a couple years ago haha

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