I've only just skimmed through this. And it's a subject I already know so I can't tell if it's actually a good resource. However, my initial impression is that I love it.
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.
Also finding things in textbooks is a real pain. It's difficult to index things in a helpful way, so you just have these counting schemes in LaTeX that increment for every definition, theorem, etc. I'd love to be able to tag things then search the tags.
And that says nothing for when you want to explain something that's difficult with static images. Being able to interact with animations by zooming, panning, pausing, slowing down, speeding up, etc. is a boon. (I don't think I actually saw an example of a non-static image here, but I think my point still stands.)
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.
I wrote an entry on Wikipedia on visualizing the QR algorithm: https://en.wikipedia.org/wiki/QR_algorithm
The visualization helped me spot an unstable fixed point and understand the behaviour of the algorithm near eigenvalue clashes. The behaviour's quite sophisticated.
I think I wrote this there, but here goes again. The idea is that a positive-definite symmetric matrix can be visualized as an ellipse. This follows from the spectral theorem. Each iteration of the QR algorithm causes the ellipse to fall towards the x-axis, as if under the influence of gravity. The unstable fixed point corresponds to when the ellipse is standing up precariously, unable to fall in either direction. If you tilt it by just a bit, it will fall over (so the fixed point is unstable).
The case when the ellipse is nearly circular (corresponding to near eigenvalue clashes) causes the ellipse to fall over slowly. I think this also makes physical sense, if you think of it being under the influence of gravity. If you think of this near-circle as being a matrix, then this matrix is nearly equal to a scalar multiple of the identity matrix, so its eigenvalues are essentially known. The fact that the ellipse falls very slowly implies that the eigenvectors are unstable near eigenvalue clashes, but the eigenvalues are easy to find.
Note: The issues surrounding the unstable fixed point can be fixed using Wilkinson shifts. This makes each iteration into a discontinuous function, allowing all the fixed points to be stable. The issue surrounding instability of the eigenvectors near eigenvalue clashes cannot be fixed, as it's intrinsic to eigendecomposition (even of symmetric matrices). The latter difficulties can be dodged by slightly perturbing the matrix, but the resulting eigenvectors can be very different from the eigenvectors of the unperturbed matrix.
I really wish we'd have these kind of tools when I took Linear Algebra (or many other math and engineering courses, for that mater).
When I took it, it was purely proofs up and down on the blackboard, zero visualizations.
Being aware of the broad applications for Linear Algebra in engineering, I'm very eager to go through some Linear Algebra course on my own. But my problem is that I can't just learn from a text, even if it's interactive. I need to apply it to something.
What are some fun projects that uses LA for an individual? I'm thinking about things like generative art, if anyone knows of any artists that inspire them.
It would be nice if the items in the left column on the index page [1] were links to the context location.
Georgia Tech! Anyone else just finish taking Graduate Algorithms? Hope you passed.
Previous discussion:
2 years ago https://news.ycombinator.com/item?id=21628449
This is great! This is a nice direction to go. Also, I don't mind if the creators would like to charge a small fee to use the book.
This is one of my favorite resources for learning about Linear Algebra. Helped me immensely when I took it last spring.
FYI, FB blocks this as "violating community standards" when you try to post it there.
And also, it's awesome!
I hope that GATech one day offers an affordable online math MSc program, like their CS program
Cool to see an article on the front page of HN from my alma mater :)
I am excited to try this, thanks for posting!
It's confusing why ml beginners are obsessed over linear algebra. The subject is very hard and need only for advanced ml model. why not just study calculus and understand it?
For those interested in these kinds of interractive math experiences, I have been keeping track of them for a while. Here is my list so far:
• https://www.intmath.com/ - Interactive Mathematics Learn math while you play with it
• http://worrydream.com/LadderOfAbstraction/ - up and down the ladder of abstraction
• https://betterexplained.com/ - Intuitive guides to various things in math
• https://www.math3ma.com/blog/matrices-probability-graphs - Viewing Matrices & Probability as Graphs
• http://immersivemath.com/ila/index.html - immersive linear alg