Show HN: A little notebook for learning linear algebra with Python
little-book-of.github.io
little-book-of.github.io
If a person has the goal of getting into a field like computer vision or machine learning, would they be able to build useful things right away if they completed this book?
I think it is actually delusional to say that a book which does not properly define what a vector is, contains the "core ideas" of linear algebra.
Linear algebra is so much more than lines in R^n. It is a powerful theory because it is abstract.
No, you aren't. How would you explain that matrices are both linear transformations and vectors? How would explain what a dual space is? How would you understand the properties of the Fourier transformation, which is a mapping between functions, which are also vectors, and itself also is a vector?
i’m as much with bourbaki as the next guy. but that’s not really how most engineers learn in practice.
as for treating linear maps between finite dimensional spaces as vectors that’s quite straightforward to do in coordinate terms.
again i refer you to the classics like Golub and van Lean that have been reprinted many many times and educated generations.
Seems like a valid definition of a vector to me. What should be added or removed, in your view?
I think that a lot of mathematics would benefit from executable code (or I would say - executable maths) - so people can play and experiment with it.
I had my own approach for teaching deep learning (https://github.com/stared/thinking-in-tensors-writing-in-pyt...), showing both mathematical formulation in LaTex, and a numeric one in PyTorch.
>A vector has:
>A symbol (we might call it v, w, or even →AB in geometry).
>Components (the individual numbers, like 2 and 3 in [2, 3]).
>An arrow picture (a geometric way to see the vector as a directed line segment).
None of these are properties of all vectors. They are properties of how vectors are represented in numpy.
At least using a theorem prover would get you closer to doing actual maths and proving.
It's probably more accurate to say courses like these teach mechanics/calculation more than they teach the theory.
This collection of lab exercises seems far better suited to the purpose than the abstract proof-based resources that some of you seem to have in mind. Fortunately, those resources are still available to those who want to go further into the theory.
And if you feel you have something to add to linear-algebra pedagogy yourself, well... what's stopping you?
That said, it doesn't lead with a notion of vector that is as general as I'd like. Readers might be later surprised to find that there are vectors which are not lists of numbers. But I only looked at the first few pages so I assume the vector space axioms show up eventually.
This is a terrible definition and makes talking about proper linear algebra proper impossible.
Writing this book has been a real challenge for me, since all the books I enjoy reading are very formal ones :))
Anyone here into Grothendieck's SGA or EGA? (Sorry, a bit off topic!)
Personally, I've always liked Bourbaki's books, but they're too formal for learning - especially in linear algebra, which I see as something meant for applications rather than pure math research.
Maybe I just oversimplified things or made them feel less "math".
It takes long hours to produce a working notebook, a synergy between concept descriptions and theorical explanations alongside code, comments of code and markdown for the theory here and there is an artistic coupled with engineering stunt you should try to accomplish yourself before criticizing. Perhaps you have, in which case a contribution as the other comment points out is the courteous thing to do.
That book must have taken months to author and is one of the best read on the subject I've ever come across.
Edit: removed incorrect fact about scalars.. they don't have a direction they have polarity.
If the latter goal outshined the former, my apologies. I love stuff like this.
I see this sort of thing as being similar to how physics is taught. Year 1: Atoms are indivisible. Year 2: Well, no, actually, we lied, they consist of elementary particles called electrons, protons, and neutrons. Year 3: Well, technically the protons and neutrons aren't indivisible either. ... Year 10: OK, fine, we have no idea. Your turn, help us figure this out.
Nobody starts rambling about quarks and gluons in grade school, and few practitioners will ever need to deal with them at all. Likewise, for most people looking to get their feet wet in ML, vectors are a 1D list of numbers, matrices are a 2D list of numbers, and tensors are lists of numbers with any number of dimensions. Definitions that are incomplete at best, but good enough to get started.
(Posted in good fun, no harm intended)
Even though this is probably introduced in a later chapter, a curious reader will be able to question why the gray dots are form with integer coefficients and leave them with a wrong impression.
I'm not really a visual guy, so let me rewatch the awesome 3Blue1Brown series and try to visualize it again:
https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
Can you please make your substantive points more respectfully? This is in the Show HN guidelines: https://news.ycombinator.com/newsguidelines.html.
Your comment would be just fine without that first sentence. When people share their work, they're putting themselves in a vulnerable position, and there's no need to lead with a smack.
NumPy is the worst way of abstracting linear algebra into a programming language. Octave is better in not obfuscating the concepts with OO boilerplate and having actually useful printed matrix output instead of ugly list representations. Even Fortran is better.
numpy is an intuitive way of doing those numerical calculations in a general use programming language. only minus is the missing gong and why commands
In practice, you'll often encounter the code before the underlying math (especially in Machine Learning), so developing intuition from both directions is always helpful.
Btw, if you enjoy pure mathematics, starting from Chapter 4 in the book, I begin defining vectors and the axioms of vector spaces:
https://little-book-of.github.io/linear-algebra/books/en-US/...