Interactive Linear Algebra (2019) (opens in new tab)

https://github.com/topics/linear-algebra?l=jupyter+notebook lists "Computational Linear Algebra for Coders" https://github.com/fastai/numerical-linear-algebra

"site:GitHub.com inurl:awesome linear algebra jupyter" lists a few awesome lists with interactive linear algebra resources: https://www.google.com/search?q=site%3Agithub.com+inurl%3Aaw...

3blue1brown's "Essence of linear algebra" playlist has some excellent tutorials with intuition-building visualizations built with manim: https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...

https://github.com/ManimCommunity/manim

microMathematics Plus[0,1,2] (FLOSS Android app)

> With microMathematics Plus, not only can you perform mathematical calculations in naturally readable form but you can also create and manage your own collection of interactive formulas!

[0] https://github.com/mkulesh/microMathematics

[1] https://f-droid.org/packages/com.mkulesh.micromath.plus

[2] https://play.google.com/store/apps/details?id=com.mkulesh.mi...

> I think that textbooks, math textbooks in particular, are an example where print publishing does a disservice. (I'm counting PDFs here too.) By having to lay everything out in print form, you have to clutter up your explanations with examples and footnotes that take up physical room. .

I agree with your sentiment. Although, I believe there is bit of problem of generalizing it to a wider field. Surely, you can make explorables with Linear Algebra, Trigonometry, Geometry, Number Theory etc, but there still exists a lot of mathematical fields, where such explanation using visualization is pretty difficult. It takes a lot of time to come up good explanations, while even longer for interactive explorables.

> All in all, I'd love to see more interactive textbooks. We've got this really expressive kind of document via the web. I think we should be taking advantage of it more.

This is something I have spent the last 4 years of my life building. I designed an alternative way of writing interactive books that takes a different approach than using just explorables. It's a conversational way of learning subjects designed for auto-didacts.

You interact (converse?) with the book and learn gradually. You can of course embed explorables into it (which I am planning to for upcoming linear algebra / discrete mathematics course), but the general process can still be generalized to any field. And if you are a LaTeX fan, you are in luck as all your notes are compiled as a Tufte-Latex Notebook. The medium is called Primer, homage to the diamond age.

You can checkout Primer here https://primerlabs.io.

A comic-based guide to understand how Primer helps you learn effectively: https://primerlabs.io/comics/introducing-primer-comics/

I think I prefer physical textbooks for many reasons I wont get in to... but I do really like the interactive and visualization elements computers offer.

I feel like teaching kids some amount of coding to be able to play with math could be very helpful.

> an example where print publishing does a disservice

What would be an example (in the area of learning) where it does not? In the age of pervasive computing printed text is just that - a printout. A screenshot, if you will. Not extremely useful. Even a basic 3D model of something (like a car engine) that you could simply rotate and zoom is immensely more informative than a static picture of the same object.

Text, such as a mathematical proof, too, could be made dynamic for the benefit of the student: you could hide or show references, insert notes, etc.

Static/printed content is merely an artifact of the past technology we continue clinging to for no good reason.

To say nothing about the environmental impact.

> I think that textbooks, math textbooks in particular, are an example where print publishing does a disservice. (I'm counting PDFs here too.) By having to lay everything out in print form, you have to clutter up your explanations with examples and footnotes that take up physical room. Here, the examples are toggle-able. If I _want_ to explore an example, I can. But I don't need to. This kind of thing is especially helpful when reviewing content for, say, a test rather than learning it for the first time.

This sounds so much like a meeting of Lamport's idea of how to structure a proof (https://lamport.azurewebsites.net/pubs/proof.pdf) and the Stacks project (https://stacks.math.columbia.edu).

Since you mentioned QR, I want to link [1] to one of my favorite articles, which explains how to use the QR decomposition to sample a random orthogonal matrix uniformly at random. The basic idea, of course, is that if A is a random matrix, and we compute A = QR, then Q is a random orthogonal matrix. But with what distribution? Well, it turns out to be almost uniform (according to Haar measure), but there are some density issues. A simple scaling of QR decomposition resolves the issue.

When I was a TA I loved giving this as an example to students. It's a great teaser for the rabbit hole that is random matrix theory.

[1] Mezzadri 2002, "How to Generate Random Matrices from the Classical Compact Groups" (http://www.ams.org/notices/200705/fea-mezzadri-web.pdf)

> a positive-definite symmetric matrix can be visualized as an ellipse.

It is clear that you mean the ellipsoid as a set of points {x | x^T * A * x = 1} (or some other constant). There is another way in which all square matrices define an ellipsoid based on how the matrix transforms a unit sphere: {A*x | x^T * x = 1} (different matrices can map to the same ellipse here, however).

I always like to clarify which one.

(I unfortunately do not have intuition for the QR algorithm, but I am distracted by the description of an ellipse falling, as if it's rolling against the x-axis)

I think the trouble with LA education is that LA is just so ubiquitous and can look very different dependent on context. No style is going to suit everyones needs.

When you took it isn't really the issue as much as where you took it. Some schools and college majors choose different ways of teaching. I liked my exposure which was over 3 courses: Calc 2 had an embedded LA course; a 2xx level computational-focused LA course (much expanded version of the 2 or 3 week Calc 2 version); a 4xx math major only proof-focused LA course.

Most students just needed the brief intro. A substantial fraction (but not the majority) got the 2xx computational form. A very small number of us took the 4xx version, and since (by then) you understood the applications there was little need to focus on computations since the interesting (to the course) bit was the higher level understanding of LA.

One possible perspective: thinking of images as a vector space, and thinking of image operations like blurring, sharpening, and even any spatial transformation as linear operators (representable as matrices).

This perspective is most straightforward with a 1-dimensional domain, like an audio signal, or the temperature along a thin rod, because there's a clear way to organize all the values in a vector, e.g. all the audio samples are just in sequence. So if that's appealing to you at all, I would recommend that first. You can do filtering (like with an "equalizer") and add effects like echo/reverb as linear operations.

I think that it is particularly fun to think of some ground-truth signal that is corrupted by an echo in a room and a microphone that attenuates some frequencies, and trying to undo that corruption.

You could try making a simple software-rendered 3D game from scratch. Basic 3D rendering is pretty much all linear algebra computations involving matrices and vectors.

Check out “eigenfaces.” It’s straight forward and works reasonably well. There are a lot of interesting applications of PCA and SVD. Signal and noise separation, machine learning, the list goes on.

Hah! I took that course with Vijay Vazirani almost a decade ago.