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Interactive Linear Algebra (gatech.edu)
980 points by kondor 12 days ago | hide | past | web | favorite | 48 comments





So far I think I prefer the visualizations in http://immersivemath.com/ila/index.html

For pure visual intuition, I prefer 3blue1brown's Essence of Linear Algebra on Youtube (https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...). Even though they're not interactive, the visualizations themselves are the most compelling and insightful.

Has anyone used both interactive texts? My initial impression is that the GA Tech text starts with matrices and applications to systems of equations (like Strang) while the immersive math one seems to start more focused on vectors and geometry with computer graphics applications.

Neither seems to have supporting exercises yet which I think really limits their use as primary texts.


I took linear algebra after college (I hold a non-technical bachelors) through a local community college.

3blue1brown is what made linear algebra really click for me. My professor also gave us some jupyter notebooks to play around with that really helped. Unfortunately he developed these himself and I don't believe they are publicly accessible.

I also purchased 'Coding the Matrix' by Philip N. Klein, which gave me another perspective on Linear Algebra. I primarily used it as an additional reference. I've been meaning to do a front to back reading of the text, but haven't gotten around to it yet. It's got some pretty decent exercises.


> So far I think I prefer the visualizations in http://immersivemath.com/ila/index.html

A thread from 2015 about that book: https://news.ycombinator.com/item?id=10183725.


> Neither seems to have supporting exercises yet

I contacted one of the authors of the book. He confirms there aren't exercises yet, but says it's a work in progress and hopes there will be in the future.


Lex Friedman interviews Gilbert Strang this week:

https://youtu.be/lEZPfmGCEk0

Gilbert’s Linear Algebra course:

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


This is based on Mathbox which is... brilliant. Here's the demo blog post that shows off what it can do: https://acko.net/blog/mathbox2/

I normally skim through these types of articles but the visualization that s/he demos is so captivating and hypnotizing

Acko is Steven Wittens[1]; no need to use a pronoun when we have a name :)

[1]https://acko.net/about


This is great! Thank you.

I’m rewriting fluid mechanics lecture notes in Jupyter (with Blender videos) and I’ve been looking for tools to visualise the maths. PyGel3D kinda works but it’s awkward and doesn’t work on nbviewer.

I’m 100% on board with the death of PowerPoint and my students seem to agree now that they’ve seen Jupyter.


This looks outstanding.

I am a math professor, and I need to choose a book next time I teach linear algebra. It may well be this one.

If anyone has in-depth experience with the book, I'd be grateful to hear about it.


It covers most of the standard material in an introductory Linear Algebra course. There is one large omission: the textbook completely ignores the notion of change of basis. There is no explanation why this was done, and a guess is that it was overlooked. The book is also very streamlined, and does not have much outside of the standard curriculum (which might be a plus).

Change of basis is covered in section 5.3 on "similarity". See in particular the subsection "Geometry of Similar Matrices".

I love how approachable the writing style is. It's very easy to write a technical introduction that assumes a lot of pre-existing knowledge of the reader. So far it's a great read for someone like me who got discouraged by notation and cloudy terms before.

Looks like it's on pair with the Book of shaders.


Link to Book of Shaders: https://thebookofshaders.com/

(And a big +1 on the recommendation)

Another piece of technical documentation that stuck out to me recently as remarkably readable (though I haven't finished reading it): https://craftinginterpreters.com/


It's insane how many math books are written assuming you already know what all the terms mean.

The Book of Shaders is half vaporware, or am I missing something?

yeah seems like they never finished it :( first half is great though


Reminds me of the Nicki Case tool:

https://ncase.me/matrix/


From https://textbooks.math.gatech.edu/ila/overview.html

> Most engineering problems, no matter how complicated, can be reduced to linear algebra.

Most problems... except a problem of too much linear algebra.


Everything is linear when plotted log-log with a fat magic marker.

Linear: having to do with lines, planes, etc.

This definition makes me worry that the book is going to operate entirely in R^n. Is there anything here about substantively different vector spaces?


Looking through the book, it seems to concentrate on R^n, and not pay much attention to abstract vector spaces. Probably a good choice for the intended audience (mostly engineers, likely).

You might prefer Hefferon's Linear Algebra, also free and open-source, and which spends more time on this.

http://joshua.smcvt.edu/linearalgebra/

Both books look truly excellent.


I've taught this course at GT a few times. For this first introduction to the material, we stay away from anything other than R^n for the most part.

Is it a real concern or are you humble bragging about the lack of advanced math in an introductory text on _linear_ algebra?

No, that's a legitimate concern. It may or may not be relevant for what you specifically care about, but it's definitely a legitimate thing to ask about.

It's not uncommon for even undergraduate linear algebra to cover abstract spaces and notions of linearity which generalize beyond R^n. For example, the function space P_n consisting of all polynomials with degree less than or equal to n. Hoffman-Kunze, Halmos, Axler and Friedberg-Insel-Spence are all examples of undergraduate textbooks which cover this material.

This isn't just theoretical. Function spaces like P_n are useful in applied mathematics. And even if you don't use function spaces, it's very common for engineers, physicists and applied mathematicians to work in the complex space C^n rather than R^n.


This is just the first semester course. In my experience teaching it at GT, it is mostly taken by freshman, and I'm not sure how many majors require it but my students have come from a huge variety of majors. I'm not sure that abstract vector spaces is that useful for, say, psychology majors. There is a second semester course that usually uses Axler.

Various function spaces were definitely brought up in my linear algebra class. That kind of material comes up even in intro differential equations (eg systems of differential equations, Lp spaces).

What are some examples of interesting interactions (that you'd like to see, not necessarily already exist) that illustrate linear algebra with non-R^n vector spaces?

Thanks to representation theory, we can model vector spaces as R^n https://en.wikipedia.org/wiki/Representation_theory


I don't really think that representation theory is required here, it's easier than that. (Although representation theory is really cool!!)

Every finite dimensional vector space over the real numbers is isomorphic to R^n, for some n. But there is not always a canonical or "unique" isomorphism. I think the real difference -- and it is a subtle one -- is that R^n always comes with an "obvious" choice of basis, and many vector spaces don't.

I think the following may be the easiest interesting example. Consider the subspace of R^3, consisting of all (x, y, z) for which

x + y + z = 0.

This is a 2-dimensional real vector space. It is isomorphic to R^2, although in some sense it does not "present as such": you have to choose an isomorphism to R^2 if you want to "treat it as R^2", and there is no single choice that stands out.

In practice, one would not necessarily construct an isomorphism to R^2, or (more or less equivalently) exhibit a basis, before working with this vector space directly.


I think you could do some interesting stuff with error-correcting codes or polynomial spaces (or more generally function spaces).

I'm not too familiar with representation theory. Do you know where I could find a worked example of constructing, say, a representation of (F_p)^m on R^n? (for prime p and m>1)


Visually, (F_p)^m can be embedded in R^m trivially, but I suspect you are interested in animations of operations on F^p, which indeed would look different from R^m due to modular arithmetic, and so would be beyond the scope of the OP.

Yes I found this offputting. It's one thing to focus on R^n, but another to give students definitions that don't even hint at greater generalisation. The follow up definition also feels very dumbed-down.

algebra: solving equations involving unknowns


Axler's Linear Algebra book is best if you want a deep understanding in more fields than R^n

I would hope that a linear algebra book only operated in R^n to introduce the concepts, and then include function spaces in order to introduce integral transforms near the end.

Abstract vector spaces is a follow up course to linear algebra at gatech

Wonderful! I remember struggling to make it through linear algebra in college because the teaching style was so unapproachable and disconnected from real applications. It wasn't until I took computer graphics that I really grasped a lot of the concepts.

Good stuff.

One point of feedback: when making a widget fullscreen and then pressing the back button, you're back at the previous chapter, rather than scrolled at the position you left in the current chapter.

I see now there's also a "make small again" button far down in the bottom right, but it's still a trap.

Probably it's solved either by having the back button close the widget, or alternatively make the widget look more like an overlay and less like a new page (plus closing with "esc" would also help)


This is really neat, and it's clear a lot of work went into the polish and interactivity. I'd love to see the elimination examples extended to show how LU factorization falls out of those steps. It seems like a key thing to grasp and lays the groundwork for understanding of related matrix factorizations like symmetric, QR and singular value decomposition.

If anyone is interested in a similar intuitive or geometric guide I open sourced my own notes and you can find them here: https://github.com/photonlines/Intuitive-Overview-of-Linear-...

I had one of the authors (Dan Margalit) as a prof when I was a math undergrad at Georgia Tech. Cool guy and a great instructor.

This extensively uses the American Institute of Math's knowls[0], which I think is excellent for math documentation.

[0] https://aimath.org/knowlepedia/


This is our (my company) take on Linear Algebra, but then with exercises:

https://app.bolster.academy/courses/chapters/en/6


Awesome for reference and understanding. Anyone know of good practice material? Or maybe thinking of a way of generating practice material that exercises the component skills needed to do/understand Linear Algebra?

I got an A- in linear algebra in college without learning anything, via multiple choice tests and pattern recognition. Recently, it's started to seem like a metaphor for the state of AI.

Is there anything like this for other areas in mathematics such as Number Theory?

Good job. Thank you very much.

Thank you



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