
The Essence of Linear Algebra [video] - espeed
https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
======
melling
Previous discussion:
[https://news.ycombinator.com/item?id=13051241](https://news.ycombinator.com/item?id=13051241)

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yamaneko
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...](https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14)

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

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

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

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

~~~
hscells
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.

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

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lottin
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.

~~~
throwaway7645
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.

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falloutx
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.

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DanAndersen
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?

~~~
afarrell
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.

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allengeorge
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.

~~~
75dvtwin
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).

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Dangeranger
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.

[https://www.patreon.com/3blue1brown](https://www.patreon.com/3blue1brown)

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afarrell
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?

~~~
afarrell
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...](https://www.youtube.com/watch?v=XkY2DOUCWMU&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&t=295)

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pzh
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.

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malikNF
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?

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

~~~
aktiur
Wow, my Saturday afternoon just disappeared.

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sixo
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.

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

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rnhmjoj
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.

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espeed
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/](https://www.youtube.com/user/DaveAckley/)

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

Socratica's series on "Abstract Algebra":
[https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...](https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-
TP6)

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

Ben Garside's series "Vector Spaces":
[https://www.youtube.com/channel/UCu5cg_Jd9XSJL_CHUskgkGw/pla...](https://www.youtube.com/channel/UCu5cg_Jd9XSJL_CHUskgkGw/playlists)

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

Numenta's ([http://numenta.com](http://numenta.com)) series "HTM School" on
the Hierarchical temporal memory
([https://en.wikipedia.org/wiki/Hierarchical_temporal_memory](https://en.wikipedia.org/wiki/Hierarchical_temporal_memory))
model for ANNs:
[https://www.youtube.com/playlist?list=PL3yXMgtrZmDqhsFQzwUC9...](https://www.youtube.com/playlist?list=PL3yXMgtrZmDqhsFQzwUC9V8MeeVOQ7eZ9)

Dan Shiffman's series "The Nature of Code: Simulating Natural Systems with
Processing":
[https://www.youtube.com/user/shiffman/playlists?shelf_id=6&s...](https://www.youtube.com/user/shiffman/playlists?shelf_id=6&sort=dd&view=50)

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

YouTube Link:
[https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...](https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-
TP6)

~~~
espeed
4th one down :)

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mathgenius
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...](https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem#Reformulations_and_generalizations)

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

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

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sndean
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](https://www.youtube.com/user/Taylorns34/videos)

Specifically their "Learning to See" series.

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Kenji
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.

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Dowwie
3Blue1Brown makes really great content

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gcoda
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

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jajool
i wish there were videos about matrix factorization methods.

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elviking
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.

~~~
espeed
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.

