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The Art of Linear Algebra [pdf]

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Re: The Art of Linear Algebra [pdf]

#13

Post by hwers »

I keep seeing the same material like this (and love it btw) but I keep thinking that it's all only scratching the surface. From what I've seen in abstract algebra this stuff goes way deeper and becomes way more beautiful. I would love the "homemade" explanations and visualizations to start enlightening us about that. E.g. so DIY machine learning folks like the amazing GAN art community that's cropping could get more tooling to pluck easy hanging fruit.

Re: The Art of Linear Algebra [pdf]

#15

Post by mathgenius »

antegamisou wrote: post #12

Also check out 3Blue1Brown's Essence of Linear Algebra https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...

And here is linear algebra as path counting [1]. This is closely related to path integrals in quantum physics. The rules for combining quantum amplitudes are the rules for combining path counts [2]. This is also how graphical linear algebra works [3]. And since we are all hackers here, if you replace the underlying number system with the min-sum semiring you get things like Dijkstra's algorithm. I really like how all these ideas are related.

[1] https://www.youtube.com/watch?v=ei6RfbplYZM

[2] https://www.feynmanlectures.caltech.edu/III_03.html

[3] https://graphicallinearalgebra.net/

Re: The Art of Linear Algebra [pdf]

#16

Post by ChicagoBoy11 »

I never understood why we don't do more of this stuff in school, and how calculus instead became the defacto advanced math curriculum in most high-schools. Students grow up working on their basic algebraic operations, solving equations, etc. Liner Algebra introduces them to the universe that lies just beyond those techniques, has very readily applicable uses, lends itself excellently to simulation/connections to computer science (which is super popular to teach now), etc.

Re: The Art of Linear Algebra [pdf]

#17

Post by macrolocal »

ChicagoBoy11 wrote: post #16

I never understood why we don't do more of this stuff in school, and how calculus instead became the defacto advanced math curriculum in most high-schools. Students grow up working on their basic algebraic operations, solving equations, etc. Liner Algebra introduces them to the universe that lies just beyond those techniques, has very readily applicable uses, lends itself excellently to simulation/connections to comp…

I suspect the answer is partly historical; without a computer, calculations in linear algebra are a pain.

Re: The Art of Linear Algebra [pdf]

#18

Post by Grustaf »

ChicagoBoy11 wrote: post #16

I never understood why we don't do more of this stuff in school, and how calculus instead became the defacto advanced math curriculum in most high-schools. Students grow up working on their basic algebraic operations, solving equations, etc. Liner Algebra introduces them to the universe that lies just beyond those techniques, has very readily applicable uses, lends itself excellently to simulation/connections to comp…

Both calculus and linear algebra are fundamental for further studies, but I'm not sure linear algebra is more approachable?

I personally find geometry and algebra more interesting, but it seems to me that derivatives are more fundamental than matrices. But maybe that's just my bias from being educated like that.

Re: The Art of Linear Algebra [pdf]

#19

Post by Grustaf »

amelius wrote: post #8

On the first page they say "if neither a or b are 0", but they haven't defined what it means for a vector to be 0. Also, they say "rank 1 matrix", but they haven't defined the concept of rank yet. Some readers might find this kind of presentation acceptable, but personally I strongly dislike it when concepts are used before they are defined.

The concept of an additive zero is built into what it means to be a vector space though.
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