[HN Gopher] The Art of Linear Algebra [pdf]
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The Art of Linear Algebra [pdf]
Author : allending
Score : 311 points
Date : 2021-09-30 08:11 UTC (14 hours ago)
(HTM) web link (raw.githubusercontent.com)
(TXT) w3m dump (raw.githubusercontent.com)
| _wldu wrote:
| This is great for those studying algorithms (DP, DC, FFT, etc.).
| Very useful.
| ivan_ah wrote:
| Wow this is really well done. It's like a visual-algebraic
| approach... I've never seen this before.
|
| I like how the author sets up a "grammar of matrix
| multiplications," and then reuses the same patterns in the rest
| of the document.
|
| For people who might not be familiar, these visual notes are
| inspired by and complement Prof. Strang's new book
| https://math.mit.edu/~gs/everyone/ and course
| https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-li...
| https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE2...
| see also https://news.ycombinator.com/item?id=23157827
| allending wrote:
| Your books are pretty amazing too Ivan. Have em all.
| visarga wrote:
| Nice, I was just reviewing SVD and PCA, I grok them but then I
| forget, this material is useful for remembering the big picture.
| [deleted]
| tishha wrote:
| good notes, very useful!
| antegamisou wrote:
| Also check out 3Blue1Brown's Essence of Linear Algebra
|
| https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...
| penguin_booze wrote:
| Here's part 1 of Pavel Grinfeld's linear algebra series: https:
| //www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1....
| mathgenius wrote:
| 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/
| [deleted]
| skytreader wrote:
| Instantly fell in love with the presentation! Might give this a
| read-through just out of pure curiosity.
|
| In undergrad, my mnemonic for these operations was visualizing
| the matrices animated in my head. The more complex ones, it was
| actually easier for me to remember Scheme functions that
| represent the algorithm (all expressed via higher-order functions
| so it was pretty concise); this was unique to my circumstances as
| an undergrad, not something I can pull off today without
| reviewing a lot of material.
|
| Presenting the operations with color and blocks just gives a more
| natural "user interface" (lacking a better term) for remembering
| it!
| Matt-Gleich wrote:
| Great visuals
| hwers wrote:
| 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.
| ChicagoBoy11 wrote:
| 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.
| macrolocal wrote:
| I suspect the answer is partly historical; without a computer,
| calculations in linear algebra are a pain.
| Grustaf wrote:
| 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.
| dls2016 wrote:
| What if I told you the derivative is a linear operator?
| (matrix meme style)
| [deleted]
| Grustaf wrote:
| Of course, but there are plenty of linear operators, that
| property alone doesn't define it.
| wbsss4412 wrote:
| This begs the question: more fundamental to what?
|
| At a high school level, you have students who are either
| engaged and will likely learn both subjects eventually, or
| students who aren't as engaged and are just looking to take
| their "last math class". In my opinion, it makes more sense
| to offer a choice or at least focus on the curriculum that
| keeps students are that age to most engaged.
|
| As someone who has studied both subjects, I'd go with linear
| algebra 9 times out of 10, the exception being if someone
| wanted to also study physics.
| SavantIdiot wrote:
| The question seems to be: should integration and
| differentiation be taught before matrix operations, or
| after?
|
| IMHO, since linear algebra is largely a tool for solving
| differential equations, I think calculus should be taught
| first, as the fundamental knowledge.
| Grustaf wrote:
| Linear algebra is incredibly useful, basically for
| anything you might want to do, differential equations is
| pretty far down the list of applications I would think
| of.
|
| But calculus is also very very useful, and probably
| easier to understand, derivatives and integral are quite
| intuitive concepts compared to eigen vectors...
| wbsss4412 wrote:
| I'm not really sure where you got the idea that it's
| largely a tool for solving differential equations.
| Certainly that is an application, but that isn't the only
| use case.
| SavantIdiot wrote:
| That's how it was introduced to me. Which is probably
| part of the discussion about what should be taught first.
| wbsss4412 wrote:
| I was introduced to the topic on its own, so that colors
| my perspective as well I suppose.
|
| If I were to think about what a high school student would
| get the most value out of, it would realistically involve
| a combination of linear algebra and statistics (getting
| into basic linear modeling/ols). You'd have to hand wave
| away some of the proofs which require knowledge of
| calculus, but high school classes aren't very rigorous
| anyways.
| ivan_ah wrote:
| There is a nice "opinion piece" by Strang that echoes what
| you're saying here: Too Much Calculus (and not enough LA)
| https://siags.siam.org/siagla/articles/Strang2001.pdf
| chobytes wrote:
| Probably historical. Modern linear algebra is extremely recent
| as math goes. Definitely much younger than calculus.
|
| Also I would add that linear algebra, calculus, and
| differential equations all go pretty much hand in hand. We
| could probably stand to teach anyone with an inclination for
| STEM all of those much sooner.
| BeetleB wrote:
| I think it's historical. Linear algebra applications fall into
| two categories:
|
| 1. Theoretical (as in vector spaces, etc). These mostly are
| useful in advanced courses in engineering/science, and a lot of
| their applications involve calculus (e.g. Fourier series,
| function spaces, etc). So calculus needs to be taught first.
|
| 2. Computational. These can be subdivided into applications
| that involve calculus (e.g. differential equations) and
| everything else (graphics, etc).
|
| Many of the latter's applications are relatively recent (last
| few decades). Whereas calculus was needed in virtually all
| types of engineering and science. So it made sense to teach
| calculus.
|
| Imagine it's the 1970's. Your in HS. What will you do with all
| the linear algebra knowledge that won't require calculus?
| Assume you have no access to computers.
| edge17 wrote:
| My guess is that it's just easier to find Calculus teachers vs
| Linear Algebra teachers. Maybe it's less obvious in the US but
| when I have visited schools in poor countries you really get a
| sense of where knowledge in certain subjects tops out. I find
| people in the US seem to assume they can magically drum up a
| supply of linear algebra teachers, or the like. STEM teachers
| are technical professionals.
| SavantIdiot wrote:
| I think highschool tries to expose you to as much as possible.
| I did get a tiny bit of linear algebra at the end of my senior
| year in calc, but back then PCs pretty much didn't exist. I
| think today LA is more applicable to programming than
| integrals, series & expansions, continuity, etc... so maybe
| times have changed.
| SamBam wrote:
| I would say that calculus is more relevant to the sciences, is
| it not?
|
| Possibly not Calculus 2, but at least getting an intuitive
| understanding of rates of change, second derivatives, and area
| under a curve seems pretty critical for all the sciences.
| amelius wrote:
| 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.
| Grustaf wrote:
| The concept of an additive zero is built into what it means to
| be a vector space though.
| Koshkin wrote:
| To be fair, it does not look like this text is intended to be a
| self-contained introduction to linear algebra.
| jnurmine wrote:
| Would love to see other things like physics, chemistry, etc. done
| with this kind of representation. Visualized, concise
| descriptions.
| jasode wrote:
| It seems like the "github.com" link would be a better and more
| canonical url rather than link an opaque url to
| "githubusercontent.com"
|
| https://github.com/kenjihiranabe/The-Art-of-Linear-Algebra
| happy-go-lucky wrote:
| And, "Graphic Notes on Gilbert Strang's Linear Algebra for
| Everyone" would be less ambiguous than "The Art of Linear
| Algebra."
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(page generated 2021-09-30 23:01 UTC)