Linear Algebra Done Right Video Series | Hacker News Reader

Linear Algebra Done Right Video Series

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196 pointsseycombi27 comments

This playlist gives a very nice geometric intuition to a lot of the foundational concepts of linear algebra:

https://m.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVF...

All videos from 3blue1brown are amazing! Followed up through the latest 'Essence of Calculus' series, really a gem.

He makes the software used to create the animations and it's open-source: https://github.com/3b1b/manim

This playlist, in conjunction with my professor's normal lectures, helped me pass my Linear Algebra class last semester. I highly recommend it!

Really good stuff from what I could tell from a glance at this! "Linear Algebra Done Right" is a terrific book, though personally I favor Lax's book: "Linear Algebra and its Applications". I have not studied Terence Tao's notes (which I obtained from AMS's link for "open math notes") https://www.ams.org/open-math-notes/omn-view-listing?listing.... Based on what I know of Tao's writing in general, this should be fun.

Note that these references have a more "abstract" viewpoint (e.g focusing on "coordinate-free" methods, such as linear operators as opposed to matries as the fundamental object) than that advocated and taught by Strang. I do not know in the end which is better for an absolute beginner. All I can guarantee is that the abstract viewpoint is definitely needed and far more useful for anyone using math beyond its basics. This includes (but is not limited to) machine learning and optimization.

I believe that the abstract viewpoint should come first, but should be tied very, very thoroughly to the concrete calculations. The construction of a matrix to represent a linear operator in two concrete coordinate systems should be completely understood, forwards and backwards.

The point of the abstraction not being to have an abstraction, but to give intuition to the otherwise pointless-seeming calculations.

I haven't read Lax's linear algebra book, but I fell in love with his functional analysis book in the summer before I began grad school. His notes on hyperbolic pde are also great. Lax is of course a great mathematician, but he is also an incredibly lucid writer. He doesn't make anything more difficult than it needs to be. I would recommend reading anything written by him that you can get your hands on.

There are three problems with Lax's book, from my experience (I used Lax in undergrad and Axler later and I would have gotten more from each if the order were reversed).

1) Lax leaves significant results to the problems, and the result is that there is little explanation of their importance.

2) Lax prefers short, mathematically impressive proofs which leverage the mathematics he has developed in the book to that point. But the proofs do not help in understanding.

3) Lax relies heavily on determinants and characteristic polynomials to prove major results. Axler deliberately avoids them, and he says why in the introduction too!

Axler and Lax have two diametrically opposed approaches to linear algebra pedagogy. Neither is "wrong" but in my experience Axler's is much better for learning and Lax's is better for a deeper understanding (and for a different approach).

Thanks a lot for this endorsement of his functional analysis book. I never got around to understanding functional analysis properly; mainly because I don't work on PDE's/control theory that would have used the ideas more heavily. Added to my todo list.

Not really on topic, but I would like to use this opportunity to express gratitude to the HN community - although this forum is mostly tech-centric, it is wonderful to see things like this post get interest and discussion.

Thank you for saying so! It's good to not feel left out. There are plenty of mathematicians and applied mathematicians here, too. We just do not comment on the JS/Rust discussions :)

> Really good stuff from what I could tell from a glance at this!

I agree. I'm really glad this was only posted on a weekend because now I'll have to spend the day watching all the videos. Awesome stuff.

I'm not a particularly advanced student, but I took an introductory class using Lax and then later material with an algebraist professor and her notes. I struggled with both, but I found it difficult to understand Lax's material until I was introduced to Galois/Invariant Theory and it all became much more clear.

With the notes I used in the latter class I felt I gained a very deep understanding in return for my efforts, especially because they went into Complexification and bilinear/sesquilinear forms which gave me a really strong intuition on the geometry of matrix representations both real and complex. If I remember correctly, Lax didn't really focus on those concepts

Not that Lax is a bad book but I feel like Lax's Linear Algebrs is actually a sneaky way of familiarizing the reader with formalisms they'll encounter in later, more abstract material, so maybe better for students who are already comfortable with a certain amount of abstraction so they're able to gain enough to leverage it later but not necessarily for a truly introductory class.

I agree with you. My teacher at Czech Technical University had a similar, abstract approach to linear algebra - also used linear operators all the way, matrices were only mentioned in passing (and in exercises).

It was a tough course for 1st year freshmen (almost as difficult as mathematical analysis), due to its abstract nature, but I was grateful for that in the end.

Oh great! So there is "Linear Algebra and Its Applications" by Lax. And "Linear Algebra and its Applications" by Lay. At least "its" is capitalized differently. ;-)

I gathered the videos into a playlist for better findability: https://www.youtube.com/playlist?list=PLF9bYy39t4tGBnJ9FjOSF...

Thanks for that!

I like this series by Gilbert Strang:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

I was recently trying to understand wireless channel estimation. So supposedly at the reciever we have to pass the signal through, an inverse channel filter to recover the transmitted signal. A friend of mine asked: what if the channel has no inverse? I had no idea how to respond to this because of my poor intuitive understanding of linear algebra.

3blue1brown's video series explained this beautifully as, when the inverse doesn't exist, the transformation packs the input into a lower dimensional space. So in communication terms the transmitted signal is completely lost anyway. So we should look for a better channel to communicate.

Now needless to say there is a lot I am probably wrong about, but I'm still very grateful for that excellent video series.

As a rogue mathematician I miss linear algebra.

I really think Gilbert Strang's lectures are responsible for creating the impression that linear algebra is hard. I watched that lecture and it was all dry math and very few intuitions. "Linear algebra done right" is a better book. Come on we are in 21st century and our intelligence has evolved so that we can understand the 16th century math better :)

We have limited attention span in 21st century. The teaching content better be interesting with low cognitive load required to understand using animations to get the material in to the brain. This is the right direction.

If you want a more advanced book I really strongly recommend Matrix Analysis and it's Applications by Carl Meyer. Very clear and concise and the problems are well written. It's my all time favorite math book. But this is really if you want to absorb a lot of more-advanced linear algebra with a lot less fluff and exposition than Strang

Does anyone have a good book for advanced linear algebra? I learned the basics in school - eg. Matrix diagonalization, svd) but I need for ml to really understand things like angle between flats, tensor decomposition, etc

It will be nice if somebody ever does a version of (I) here's the theory behind a linear algebra equation, and then, (ii) here's a real world application.

First, math (linear algebra in particular) is not at all about equations. The format you said is very limiting, because most of the important things in linear algebra are not equations: vector spaces, basis, norms, orthogonality, diagonalization, eigenvalues, etc.

Second, if you crave for applications, check out this free linear algebra book: Linear Algebra Done Wrong by Serge Treil (https://www.math.brown.edu/~treil/papers/LADW/LADW.html)

His book has a lot of applications, and has been written as a response to Axler's book, which does not (even though I think that Axler's way is the best).

Finally, No Bullshit Guide to Linear Algebra tries to do exactly what you want: the presentation aims to be

(I)here's a concept (II)here's a real-world application

The book is here: https://gumroad.com/l/noBSLA, $29

(I found out about this book on Hackernews, which seems to be a pretty decent place for such things)

Thanks for he links. I'll check them out. No Bullshit guide seems interesting.

In my expereince, nothing beats practice.

If you want to play the game, first you need to learn the rules.

Theory beats practice, because without theory there is no practice.

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