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Comment by tptacek | Hacker News Reader

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just have it.

I learned from Strang, for what it's worth, which is basically LU, spaces, QR, then spectral.

I am really bad at math, for what it's worth; this is just the one advanced math subject that intuitively clicked for me.

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MIT OCW is an amazing resource -- anyone can learn from Strang, which is a goldmine.

He also created a course on using Linear Algebra for machine learning:

> Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization–and above all a full explanation of deep learning.

- MIT OCW Course: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-an...)

- The text book website: Linear Algebra and Learning from Data (2019) https://math.mit.edu/~gs/learningfromdata/

- The Classic Linear Algebra Course: https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010...

> I didn't think any part of linear algebra was boring.

My formal linear algebra course was boring as hell, to me. The ~4 lectures my security prof dedicated to explaining just enough to do some RSA was absolutely incredible. I would pay lots of money for a hands-on what-linalg-is-useful-for course with practical examples like that.

Try this. :)

https://kel.bz/post/lll/

(If you work through the prerequisites and use "understanding this post" as a sort of roadmap of what you actually need to know, this gets you about 2/3rds through undergraduate linear algebra, and you can skim through nullspaces --- all in the service of learning a generally useful tool for attacking cryptosystems).

Thanks for the notes. This is marvellous. I do not work on, or have interest in cryptography algorithms, but this is such an interesting read.

Kelby Ludwig is such a talented explainer, it upsets me.

There's a book called ILA (Interactive Linear Algebra) that I found really good: https://textbooks.math.gatech.edu/ila/

I haven't looked at Strang's approach.

The "x = b / A" is a bit of a gut-punch on first look because my mind immediately tells me all the ways that that does not work. It makes a some sense once I take a second to think about it, and I can see why it would make you want to jump in a little deeper, but matrices being non-commutative makes me cringe at the idea of a division operator which does not very very clearly spell out where it appears in the chain.

Ax = b is all well and good, but AxA^-1 = bA^-1 is not meaningful; the application/composition order is very important.

Didn't know about Gaussian elimination until today. Really cool, and really useful! Thanks for sharing!

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