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Comment by zvmaz | Hacker News Reader

Linear Algebra Done right did not help me try to demystify linear algebra. Quite the contrary. It is the Youtube channel 3Blue1Brown[1] that gave me an intuition of what linear algebra was about, and I am forever thankful to him.

Watch it and if you are a teacher, don't be a smug "formalist".

[1] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...

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I think a good project-based way to learn about linear algebra is to build a 3d scene rasterizer from scratch. Once you get nested transforms and projection matrices figured out, you are probably going to have a much deeper sense for what these things actually are than someone walking out of a typical college course. The 4x4 homogenous transformation matrix sort of tricked me into learning about things I didn't originally intend to. I spent a solid week watching videos (such as the exact one you linked above) to reach a more satisfying understanding.

I vividly recall manipulating matrices on paper for weeks in college. Absolutely none of it had any real meaning to me. Very little of that education helped me as I got into actual projects, other than some vague awareness that there are these things called matrices and something about system of equations. Having something to pull me towards a specific objective has always resulted in a much more substantial education. Learning about some of this stuff in isolation is really quite painful. Learning about painful things to get to the fun bits seems more tenable.

The homogeneous thing is very nice. As you know it let's you pack translations and self linear maps in K^n into the linear maps of K^(n+1). OpenGL uses it. Also if your perspective is isometric (the projection of the 3D canonic base reminds a Mercedes-Benz logo) it is just linear K^3->K^2.

> I think a good project-based way to learn about linear algebra is to build a 3d scene rasterizer from scratch. Once you get nested transforms and projection matrices figured out, you are probably going to have a much deeper sense for what these things actually are than someone walking out of a typical college cours

I don't think this is the best way to learn about linear algebra. It's still mystifying.

Logged in to comment this. Very much agree. It's crazy (to me) that books like this gets so much hype on HN when there are much better videos that explain these topics better.

Most of my math learning starting in high school all the way through undergrad was done by watching YouTube videos. I used books to practice problems, but when it came to understanding topics more deeply, it was always some random person on YouTube who did it better.

I hope in the future, all math (at least applied math) is explained using nice visualizations + videos instead of books like this.

The same thing applies to computer science. Try figuring out even the basics, like merge or quick sort, using pseudocode in a traditional algorithms book. It's an extremely difficult and time consuming nightmare. But watch a video of how merge or quick sort works, then you gain geniune understanding within minutes.

Surprisingly one of the best summaries (~10 pages) to applied linear algebra I've found is in Nielsen and Chuang's Quantum Computation and Quantum Information.

Presented primarily without proofs which whilst argubably can be limiting isn't relevant, at least for what their goal is.

For what it’s worth, Axler’s book did help me considerably.

Formalism isn't really the problem though. The problem is that it needs to be accompanied with geometrical explanations and intuition (especially for linear algebra).

Formalism is here to help you put words on intuition. Intuition without formalism is as useless as formalism without intuition.

In linear algebra in particular, people who avoid formalism at all cost tend to focus on obscure calculation results on matrices instead on their more geometrical counterparts on linear maps

Different audiences. 3B1B can be very effective for intuition. It does not, however, prepare you for proofs or anything rigorous or thinking about things in an axiomatic way. For example, it isn't going to provide a basis for anything infinite dimensional.

Why do so many people use the word intuition when they mean understanding?

It's an ancient and likely irrevocable error, like "font" instead of "typeface" (no really, go look it up, we have all been saying "font" when the thing we're talking about is (well, was) "typeface". I only know this because Woz used to tilt against this particular windmill.)

Jef Raskin pointed out years ago in his "Humane Interface" that in UI when we say "intuitive" we really mean just "familiar".

His example was the computer mouse. He gave one to an architect (buildings not software) friend and they turned it upside down and used it like a little trackball with their fingertip. (Raskin is that old mice were new.) Few read or heed Raskin.

Intuition is a vehicle for understanding. "Determinant says how much the matrix stretches things if treated like a function" is an intuition but doesn't give full understanding of the determinant.

Because I still don't pretend to fully understand linear algebra. But it's not a mystification anymore, thus "intuition".

"Intuition is the ability to acquire knowledge, without recourse to conscious reasoning or needing an explanation." https://en.wikipedia.org/wiki/Intuition

I don't think this is what you meant to say - have a partial understanding of something is not the same thing as intuition which is more like a "gut feeling".

> intuition which is more like a "gut feeling".

That's what I meant.

I don't know whether approaching linear algebra using a "gut feeling" approach is a good idea though.

thanks for this mate

3B1Bs Essence of Linear Algebra is great, whatever this pdf is is not

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