Interactive Linear Algebra

https://textbooks.math.gatech.edu/ila/

by kondor • 7 years ago

981 49 7 years ago

textbooks.math.gatech.edu

melling• 7 years ago

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...

Zhyl• 7 years ago

Reminds me of the Nicki Case tool:

https://ncase.me/matrix/

bhntr3• 7 years ago

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.

sh-run• 7 years ago

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.

dang• 7 years ago

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.

impendia4• 7 years ago

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.

tiborsaas• 7 years ago

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.

viburnum• 7 years ago

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

paulgb• 7 years ago

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/

newnewpdro• 7 years ago

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

percentcer• 7 years ago

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

gautamcgoel• 7 years ago

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.

mamazaco• 7 years ago

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

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

kd0amg2• 7 years ago

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?

impendia4• 7 years ago

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.

gowld2• 7 years ago

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

impendia4• 7 years ago

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.

kd0amg2• 7 years ago

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)

ajkjk• 7 years ago

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.

nostrebored• 7 years ago

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

vcxy2• 7 years ago

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.

sitkack• 7 years ago

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

throwawaymath• 7 years ago

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.

chobytes• 7 years ago

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).

zelly• 7 years ago

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

Myrmornis• 7 years ago

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

impendia4• 7 years ago

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.

qqaazz2• 7 years ago

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).

cbolton• 7 years ago

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

thinkingkong• 7 years ago

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/

syntaxing• 7 years ago

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

romwell• 7 years ago

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

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

knolan• 7 years ago

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.

kernoble• 7 years ago

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?

krosaen• 7 years ago

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.

jacobsimon• 7 years ago

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.

majke• 7 years ago

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.

black6• 7 years ago

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

lotaezenwa• 7 years ago

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

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

Aardwolf• 7 years ago

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)

perl4ever• 7 years ago

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.

iLemming• 7 years ago

Also great resource: https://aiprobook.com/numerical-linear-algebra-for-programme...

Jahak• 7 years ago

Good job. Thank you very much.

s3arch• 7 years ago

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

x7k• 7 years ago

Thank you

photon_lines• 7 years ago

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-...

Interactive Linear Algebra

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