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As a math major, I scored a perfect 100 on my Linear Algebra exam in 1974. However, just two days later, I couldn't recall a single thing.

A few years ago, with ample free time, I decided to refresh my (nonexistent) memory by watching online linear algebra lectures from various professors. I was surprised by their poor quality. They lacked motivation and intuition. Khan Academy offered no improvement. Then, someone recommended Linear Algebra Done Right (LADR). I read it three times, and by the third iteration, I finally began to appreciate the beauty of the theory. Linear algebra is a purely algebraic theory; visual aids are of limited help. In short, if you have the time, I recommend reading LADR. Otherwise, don't bother.

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I don't know whether LADR is good for someone who is new to linear algebra. I've seen it recommended so many times, so ~12 years ago when I was living in Beijing I bought two copies (one in English for me, and one in Chinese in case I needed to ask a colleague for help).

It took me time to study each page, to understand the examples, and then to attempt the exercises. It seemed very beautiful.

Then one day I came to a part I couldn't understand: I didn't see how something Axler said followed from the earlier stuff on the page actually followed. I scratched my head for a couple of hours, which is much longer than I'd spent on any previous page.

Eventually I asked a colleague for help. I showed him the page. He asked me to explain what I didn't understand. I started to explain what I knew, and how I didn't understand how this thing followed. As I was explaining it, that part suddenly clicked.

But I got stuck a few more times and didn't persevere.

I wonder whether it would have been better for me to have studied some numerical approach to linear algebra (like Strang's videos) first, rather than going straight into a book that's so abstract and proof-based.

I suppose it depends on your mathematical background.

(Your comment made me think about those folks who were once fit and muscular, then years later they are out of shape, and then they decide to get in shape say how easy it was to get back in shape. They don't realize that part of what made it easy is that they were once in shape, and they still more muscle cells or whatever.)

Very true. But the same applies to teaching. Mathematicians don't know where even to begin - for some of them, it's all too obvious. But the same happens with any subject. Someone proposes a certain design - but after many (20... 30... 40) years in business, you feel the design won't ever work, and try to explain, and fail because you don't know where to begin.

I got a B in my linear algebra course which was basically only numerical. I’d have gotten an A but the professor thought mountains of homework was teaching and I refused to do it all. Suffice it to say I aced every test and all the homework I actually did. None of it helped in understanding and like the grandparent I remembered none of it at the end and turned to LADR.

I don’t think any of that numerical approach helped when I read LADR. LADR isn’t about “doing the work” it’s about “doing the work to understand”. Similar to your experience I remember reading the first chapter and then among the first chapter questions I saw questions that looked like they had no basis whatsoever in what I thought I had just learned. Then, eventually, it clicked. That’s, frankly, the only way it works with Axler, so if you want it, you’ve got to do it.

My advice is to not waste time with the numerical approach and just do it.

I had a professor who used to say “being a student is suffering” but he used it to justify a bunch of bullshit. In this case, though, I’d agree with him. LADR is suffering d followed by satisfaction (and rinse and repeat).

For me, the main solution was to apply it to another problem that uses Linear Algebra as Application, which in my case was Introductory Quantum Course and implementing BLAS using Rust and C. That way you keep thinking and using this info. Otherwise, information in vacuum seems to abstract to care about.

I really like your approach. Any resource recommendations?

On what topic? Linear Algebra, Quantum Mechanics?

Linear algebra applications resources. Something that I can follow up with examples.

For me, it was introductory quantum mechanics (QM) books, you can go with MIT online course from Barton Zwiebach and online course from BLIS (This is for Rust/C implementation of BLAS). If you fall in love with QM and go for more rigorous formulation of its mathematical structure, you can follow it up with An Introduction to Hilbert Space by N. Young, which was the book used in my next semester for Hilbert Space Course.

Hilbert Space is the mathematical framework to describe QM systems.

You might enjoy "Thirty-three Miniatures" (2010) by Jiřì Matoušek. It's a collection of short applications of Linear Algebra in geometry, combinatorics and CS.

https://kam.mff.cuni.cz/~matousek/stml-53-matousek-1.pdf

Have you tried Math Academy? I think the difference is that it's actually made by a team of mathematicians creating the content manually.

Not yet, but now I will, just out of curiosity. There's a problem with mathematicians teaching the subject. After all, the youtube lectures were also given by mathematicians. In attempt to make things "accessible", they de-emphasize the algebraic part of the subject and replace it with... I don't know what. The common theme is to consider only R^n. That's not what it's about. Maybe Math Academy course is different though.

That's not a "mathematician" thing, it's a US thing. US universities, for some reason, insist on teaching mathematics twice, once with lots of handwaving and then at some point you get to do a "proof-based course".

In Europe (at least in certain countries, can't speak to all of them), maths lectures will typically be abstract and proof-based from day 1 - at least for maths majors (but frequently for CS and physics students too). Other majors, such as economics and maybe engineering, may get their own lectures that tend to be more hand-wavey because they don't necessarily need the axioms of real numbers to take a derivative here and there.

My linear algebra course was algebra and proof based to the extent that maybe a little bit more geometric intuition would have helped.

3blue1brown is pretty good.

3blue1brown's linear algebra series is very different from what GP is talking about.

If you think linear algebra is something geometric, like "a 3x3 transform matrix is rotation and scaling; an eigenvector is something after transformation and parallel to its old self..." you will be surprised at how little LADR talks about these.

On the contrary, the most important part (imo) of 3b1b is that it helps you intuitively get these geometric interpretations.

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