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[dupe] The Essence of Linear Algebra [video] (youtube.com)
440 points by espeed 40 days ago | hide | past | web | 41 comments | favorite

3Blue1Brown explanation of eigenvectors and eigenvalues is very insightful and intuitive [1]. One of my favorite videos from his channel is "Who cares about topology? (Inscribed rectangle problem)" [2]. They way a Torus and Möbius Strip come up as solution to the problem is so elegant.

[1]: https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQ...

[2]: https://www.youtube.com/watch?v=AmgkSdhK4K8

There's a new topology video out! As cool as the first one!

Here is the animation engine used in the videos (written in Python):

Manim: animation engine for explanatory math videos https://github.com/3b1b/manim

I always thought he used processing. The fact that he built all the animations with his own library makes his videos way more impressive now.

He jokes about it in the series. That he used linear algebra for it too.

The introductory video is absolutely right. I studied linear algebra at university but it's been only recently when I decided to re-learn it that I have finally got it, and for that understanding the geometry behind linear algebra is crucial. Now I find myself thinking about problems in terms of matrices and vectors and making all sorts of deductions, even writing mathematical proofs, which I never thought I would.

Sounds like you'd enjoy playing around with array programming languages like APL (dyalog, microapl), J, or Q w/ kdb+(kx systems). As one of the oldest paradigms, there are a few open source and proprietary-commercial offerings.

This is probably the best Math Channel on Youtube. I highly recommend watching his Topology videos. Numberphile, Mathloger and Vsauce are other great youtube Channels which I subscribe to.

These videos are a treasure and I watch every new video that 3Blue1Brown puts out. They saved me last year when I had a graduate-level numerical linear algebra class and was struggling to grok the true meaning of all this linear algebra stuff (an embarrassing situation for a computer-graphics guy to be in!). Things that had always been "insert formula X to get result Y" started making a lot more sense.

The videos also make me angry because it frustrates me that such explanations were not available to me earlier in my life. What is it about the state of math education that this kind of explanation is not there in every class?

I think the problem is that until recently, the costs of producing and distributing these videos was much higher. With the rise of YouTube, it has become possible for a Salman Khan or John Green to become a celebrity eduvideographer without a lot of capital investment.

One of the flaws of common core is that it seems that the proponents did a poor job of marketing it to the broader adult community. Doing so would have:

1) Given parents the answer to "why is this change happening at all? I learned math just fine as a kid!"

2) Engaged some people in trying to think about the best ways to explain that material and accelerated the formation of a community that gains status with each other by coming up with better and better explanations of the common core curriculum.

What is it about the state of math education that this kind of explanation is not there in every class?

Computational skill is used as a proxy for understanding because educators, just like the rest of us, are lazy-ass human beings. The authors of standardized tests don't care if you know what an eigenvalue is, only that you can calculate one.

Or, put more charitably: teaching math is hard as hell.

Like so many other aspects of our lives, modern math education in the US grew out of a knee-jerk response to a perceived crisis -- the launch of Sputnik, in this case -- that wasn't very well thought out. Schools were required to measure kids' progress in math and science quantitatively, precisely, and repeatably. And just as in other fields, once a measurement becomes a target, it ceases to be a good measurement.

I have to echo what the posters above have already said: I learned linear algebra way back in the day, and it's only now, watching the videos, that I suddenly _got_ matrix multiplication etc. I'm not joking when I say I almost cried with joy. These videos are incredible, and I'm so sad I didn't see them earlier in my life.

Same here, felt like 'crying with joy'. I have been struggling to really understand and internalize eigen vectors for years now (even I used them for some of my lab work). With this visualization, I felt like it had opened another, previously closed, door for me. I would like to donate to this author (and octave's author, on a separate note).

3blue1brown's videos are the best I've found for linear algebra and mathematics in general, they are excellent and insightful.

If you value this kind of material please consider supporting via their Patreon page.


These explainations are fantastic! Does anyone know of a literature class that examines different technical explainations and analyses why they are successful and where they fail?

So one thing I just noticed: in Video 4 (Matrix Multiplication) when he shifts from just showing geometry to a walkthrough of the numerical operations he begins tracing the paths of i-hat and j-hat. This continues to keep the explanation concrete. In fact, it is almost as if he is a programmer debugging a set of functions and walking the path of a piece of data from through one function call to another.

You can follow this along here: https://www.youtube.com/watch?v=XkY2DOUCWMU&list=PLZHQObOWTQ...

Bret Victor's essays, demos, and talks focus on this topic:


I'd also heartily recommend Gilbert Strang's Linear Algebra video lectures (OCW MIT). They seem to have the same goal--to develop a very strong geometric intuition.

Wow thanks for posting this link. I was wondering does anyone know more channels (or any other resource) just like this one, for Statistics and Calculus?

These are fun too, not directly calculus but some infinities and series videos https://www.youtube.com/channel/UCOGeU-1Fig3rrDjhm9Zs_wg

Wow, my Saturday afternoon just disappeared.

One of the best channels that I've found for Calculus is:


He also has a Statistics series, as well as some pre-calculus / basic algebra stuff.

This is an amazing teacher, great channel for learning calculus: https://www.youtube.com/channel/UCoHhuummRZaIVX7bD4t2czg

Check out Mooculus series on YouTube

There are great, like everyone 3b1b makes. I'm comfortable with lin alg but I'm skimming them to see what he's chosen to emphasize. I love - for example - that he points out how "span" becomes more interesting in 3d, obv with an example, and the underemphasis on the mechanical operation of matrix multiplication relative to the geometric one.

I do wish the formulas for the dot product and determinant were derived from the geometric explanation, rather than justified with it afterwards. I always appreciated that in classes.

There are some more advanced topics in lin alg that I would have loved to see get the full visual intuitive treatment when I was learning these things.

- SVD, because it's more general and less pathological than eigenvalue decomposition, and often more useful. - A linear transformation as consisting of (I think, it's been a little while) a choice of eigenvectors and eigenvalues, "divided" by the extra degrees of freedom from duplicate eigenvalues. - The "taxonomy" of normal matrices and the polar decomposition (obviously comes after complex matrices)

And there's a nice visualization of the mechanical algorithm of matrix multiplication that looks way more "plausible" than the normal one: draw your two input matrices and your output matrix on grids on 3 sides of a rectangular prism around a corner. Then each value in the output matrix is the dot product of the vectors that intersect at that coordinate, and the whole thing only looks right if all the dimensions match up correctly.

I like how the intro music of each video relaxes me into the subject.

I took a Linear Algebra course last year: the relationship between linear transformations, matrices and basis was carefully explained and also given a clear geometrical interpretation but the determinant was introduced as some magical matrix function with all the property we needed (multilinear, alternating, ...) and the just proceeded to prove its existance and uniqueness. I have never even thought it was related to area before. Also the justification (in chapter 8 part 2) for the formal determinant calculation to obtain the cross product is amazing.

I'm still not sure what the essence of the cross product is: how it is related to divisions algebra (quaternions and octonions) and how bivectors fit in the context of general vector spaces. This was not in the scope of the course however.

Other good STEM video channels/series:

Dave Ackley's series "Hyperspace Academy", "Robust First Computing", and "Artificial Life" are well animated: https://www.youtube.com/user/DaveAckley/

Vsause's video on "The Banach–Tarski Paradox": https://www.youtube.com/watch?v=s86-Z-CbaHA (mindblown)

XylyXylyX channel's series on "What is a Tensor?" and "What is a Manifold?": https://www.youtube.com/user/XylyXylyX/playlists

Socratica's series on "Abstract Algebra": https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...

Mathologer's channel on all-things math: https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg

Ben Garside's series "Vector Spaces": https://www.youtube.com/channel/UCu5cg_Jd9XSJL_CHUskgkGw/pla...

PatrickJMT's channel on calculus, game theory...(too many to list): https://www.youtube.com/user/patrickJMT/playlists

Numenta's (http://numenta.com) series "HTM School" on the Hierarchical temporal memory (https://en.wikipedia.org/wiki/Hierarchical_temporal_memory) model for ANNs: https://www.youtube.com/playlist?list=PL3yXMgtrZmDqhsFQzwUC9...

Dan Shiffman's series "The Nature of Code: Simulating Natural Systems with Processing": https://www.youtube.com/user/shiffman/playlists?shelf_id=6&s...

Thanks a lot for sharing. Have you checked out Socratica's Abstract Algebra series?

YouTube Link: https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...

4th one down :)

Thanks a lot for this nice list.

I'm just starting to realize that there are whole other levels of depth of understanding underneath this geometric intuition. So if you manage to grasp what is in these videos, don't stop! It keeps getting deeper and more mysterious, [1][2].

[1] https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem#R...

[2] http://math.ucr.edu/home/baez/week233.html

Wow, thank you so much for posting this link. I was looking for something like this before jumping into theoretical machine learning.

It can be pretty hard to sift through, but there's some really good content on YouTube. WelchLabs is another great channel: https://www.youtube.com/user/Taylorns34/videos

Specifically their "Learning to See" series.

Oh wow, I wish I would have watched that 6 years ago. But maybe it only makes sense now because I already learned it all.

3Blue1Brown makes really great content

I feel like this is changed my life just now. Even functional programming will be more meaningful with this. I want to learn haskell now

i wish there were videos about matrix factorization methods.

I went to a top school in the US and I got a major in CS without ever taking linear algebra, which in hindsight seems completely crazy. Every major in science or eng. should have this as part of the mandatory curriculum.

Hmm...it's been core curriculum in my school's CS department since I was there (almost 20 years ago) -- this is the first time I've seen visualizations like this though.

The next evolution will be to make these type of video lectures/visualizations interactive by implementing the math animation engine in JavaScript/ClojureScript and syncing it with the audio.

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