The Little Book of Linear Algebra

Posted by scapbi 9/2/2025

The Little Book of Linear Algebra(github.com)

476 points | 129 commentspage 2

ddavis 9/2/2025|

The organization and formatting of the single .tex file is such that one could almost read the source alone. Really nice. Also, I had no idea that GitHub did such a good job rendering the LaTeX math in markdown, it's imperfect but definitely good.

jean_lannes 9/3/2025||

As someone who took a standard undergrad linear algebra course but never really used it in my work, what are some good ways to get acquainted with practical applications of linear algebra?

defrost 9/3/2025||

There were some hints upstream: https://news.ycombinator.com/item?id=45107638

Machine learning, LLMs, RSA, etc.

It's generally useful for multivariate statistics, 3D flies (insects), in 3D space, clustering about a narrow slanting plane of light from a window slit are points that can be projected onto "the plane of best fit" - nominally the slanting plane of light.

That right there is a geometric picture of fitting a line, a plane, a lower order manifold, to a higher order data set, the errors (distance from plane), etc. and something of what Singular Value Decomposition is about (used for image enhancement, sharpening fuzzy data, etc).

The real test of applications is what kind of work do you see yourself doing? - A quick back read suggests your currently a CS student, so all unfocused potential for now (perhaps).

JBits 9/3/2025||

A good use of linear algebra that I'm working with at the moment is the use of splines as a basis for real (vector) functions. After obtaining the matrix/vector representations you can solve for the spline coefficients (and then plot them).

Linear transforms (such as rotations and displacements) in GPU graphics.

Fourier series in signal processing.

JPEG compression.

Obtaining the best fit element in a vector space of curves given data or other constraints.

Understanding autodiff in JAX.

The mathematical definition of a tensor helps develop intuition for manipulating arrays/tensors in array libraries.

Transition matrices of a Markov chain.

PageRank.

russellbeattie 9/2/2025||

Someone should convert all the examples into C code so it's more intelligible to programmers who are, let's admit, the main audience for something like this.

To the best of my knowledge: Scalars are variables. Vectors are arrays. Matrices are multi dimensional arrays. Addition and multiplication is iteration with operators. Combinations are concatenation. The rest like dot products or norms are just specialized functions.

But it'd be nice to see it all coded up. It wouldn't be as concise, but it'd be readable.

CamperBob2 9/2/2025|

That's basically what you'll get if you pick up a book on 3D game programming. However, progress will come to a halt when you get to things like determinants and eigenvalues that don't show up in core 3D graphics pipelines. You'll have to find other ways to motivate a C version of that part of the curriculum... but I agree, that's a well-worthwhile thing to ask for.

eliaskickbush 9/2/2025||

Highly recommend to anyone struggling with linear algebra to check out Linear Algebra Done Right, by Sheldon Axler. Do always keep in mind that some concepts are very verbose, but truly out of necessity. If you're talking about an N by N matrix, you're naturally going to have to distinguish N^2 different elements.

You can go very far without touching matrices, and actually find motivation on this abstract base before learning how it interops with matrices.

akww 9/3/2025|

A little surprising to me this doesn’t come up the most! Excellent text, and look, you can get the 4th edition (2024) for free as a pdf at http://axler.net.

ivan_ah 9/2/2025||

Wow very nice. Lots of content in here, with no lengthy explanations but useful point-form intuition.

The .epub has very clean math done in HTML (no images), which is a cool way to do things. I've never seen this before. I wonder what the author used to produce the .epub from the .tex?

ivan_ah 9/2/2025|

Update: using Sigil to look inside the .epub, I saw it was produced by `pandoc` and the math is rendered as MathML.

defanor 9/2/2025||

Always nice to see CC-licensed textbooks. This one looks fairly minimal, not including much of explanation, illustrations, or proofs; I think those are generally useful for the initial study, but this should still work as a cheat sheet, at least.

bobajeff 9/3/2025||

I love this. It's a great companion to all the resources online like Kahn Academy, 3blue1brown etc. and mathisfun, Wolfram Mathworld and Google's Gemini.

Edit: Also the mathematics stackexchange.

anthk 9/2/2025||

Klong/k looks ideal for this to have some practice with linear alg concepts:

http://t3x.org/klong/index.html

rramadass 9/3/2025|

This was one of the reasons for me wanting to study the J programming language. I have a notion AI/ML programming might be better done using array languages.

Some books for studying Mathematics using J are listed here - https://code.jsoftware.com/wiki/Books

vismit2000 9/3/2025|

Related: Immersive Linear Algebra - https://immersivemath.com/ila/

More comments...