What is...the exterior algebra?
HTML-код
- Опубликовано: 4 июл 2024
- Goal.
Explaining basic concepts of linear algebra in an intuitive way.
This time.
What is...the exterior algebra? Or: Anticommuting polynomials.
Slides.
www.dtubbenhauer.com/youtube.html
#linearalgebra
#algebra
#mathematics
Superb! When a mathematical object (like exterior algebra) seems unmotivated, it's because years of "polishing" have gone into removing all traces of the origin of the object from what the properties of the object are.
But when you bring back the motivation, or how the discoverer of the object, thought of the object, then it doesn't seem that mysterious anymore! Thanks for this wonderful content.
Thank you very much for your kind words!
And yes, that is kind of how I feel as well: I never “really understood” the exterior algebra (I formally understood it, sure, but that is very far away from understanding it ;-)) until someone told me were it came from.
By the way, it is not necessarily bad to ignore the origin. Having a clean definition is certainly important, and sometimes the original motivations and formulations were messy and the original texts are not really readable from a modern point of view. Decades of rewriting and rediscovering mathematics has an influence, and most often a positive one. But not always, which was the point of this video ;-)
You, along with Aleph 0 and Norman Wildberger, are the best Math communicators on youtube. The 'Bright Side of mathematics' channel is a close second, while Fred Schuller's lectures on youtube are the best for theoretical physics bar none--with Susskind a rather distant second. How much shall we bet, that once you upload Algebraic Geometry series, your channel is going to deservedly blow up? I am talking 100k+ within probably 6 months max. of uploading them. I promise, you make an Algebraic Geometry series, that includes sheaves, schemes, stacks and topos theory, you'll be set for life. Because NOBODY ON RUclips HAS THUSFAR SUCCEEDED in communicating it at the proper level (mathematically rigorous, while respecting the audience's need for intuition and an overarching vision of the topic)! That's where you come in!
I have thought to myself for a while, actually, why you're so good. Your production values are rather amateur (compared to the likes of 3blue1brown and others), you don't have the fancy guests or juvenile humor like numberphile, and you don't have the eloquence and authoritative mystique that some other communicators have. But maybe this is exactly what makes your channel so good.
You start basically each video the same way: abstract mathematical concept that seems intimidating, and you rephrase it in a rather easy to understand and visualize way at the bottom. Like here, Top: Exterior Algebra. Bottom: anti-commuting polynomials. Aha! That we can understand. And then you proceed to provide some basic intuition and examples, explaining (or reminding) all main concepts along the way (which I can't stand about other channels failing to do so!), along with the formal definition and how it relates back to the examples, which strengthens the intuition and formal understanding of course.
No fancy computer animations needed. No fancy guests jumping up and down and acting like fools, or overlong, drawn out, eloquent speeches that ultimately don't lead anywhere (yes I am talking about you Eric Weinstein). And importantly, you don't test the viewer's patience by sitting there hand-writing every god-damned definition, while I am sitting here growing greyer hairs on my beard. You have the slides ready, but fill them out appropriately as you go along, filling in whatever gaps or questions there may be. It is almost as if you know what questions your audience might have, and proceed to answer them in advance, without insulting their intelligence by being pedantic.
This mix is almost impossible to find anywhere else on youtube. And believe me, I have looked far and wide. Like I said, Aleph 0 and Norman Wildberger are also excellent in their own ways, but Alpeh 0 takes centuries to upload each video, and his latest on Algebraic Geometry left a lot to be desired (as usual--a notoriously difficult subject to communicate properly), and Wildberger trips over his own pedantic obsession with keeping everything "finite" and "rational." It's become a rather annoying fetish of his.
So these are my 4 and a half cents on why you should keep doing what you do, and I prophesize that your channel will finally get the wide-range audience it deserves, once you tackle Algebraic Geometry! Well done, and keep going!
p.s. My wish list for future videos:
-Algebraic Geometry (of course)
-Model Theory (nothing too fancy. Just some axiomatic set theory, some basics of Turing machines, ZFC axioms, Gödel Incompleteness, culminating in Cohen Forcing).
-Differential Geometry (yes, there is plenty of decent material out there, but I am very curious about your take on bundle theory and differential forms in general.) At least if you can include "What are bundles" and "What are differential forms" in "my fav theorems" section...I just would love to see your take on them, and what their advantages are. Also the Hodge Star Operator would be good. Cheers
Haha, thanks for the gentle reminder. I am on it 👍
Having watched quite a few videos on the topic, I can confidently say none explains it as well as this one. I was happily surprised to also get a better understanding of the determinant as part of this video. Thanks a lot!
I am flattered, but I do not deserve your praise.
I try to explain math the way I think about it, which is a very biased perspective, of course. There is no correct way of explaining it. So what you did is the best approach: get as many different points of view as possible. (This is a strongly recommended approach!)
Anyway, seems like the video was reasonably helpful and you feel the same as I do about exterior algebras, which makes me very happy.
Amazing explanation! Perfect blend of intuition and details. Now I am going to check out your other videos and understand those words from math which have always scared me :)
Thank you so much for your kind words, I am flattered.
I try to explain math the way I like to think about it, which of course might not work for everyone. It is always good to know that someone feels the same. I hope you also enjoy some of the other videos.
Thank you so much! It helped me understand the intuitive motivation behind the concept of exterior algebra.
You are very welcome! The explanation worked well for me, and I am happy to hear that it worked for you as well!
On the first slide, I have problem understanding the notations in: R[X1, X2, X3] = R/[XiXj = XjXi]. I suppose that R[X1, X2, X3] is the set of all polynomials in variables X1, X2, X3 with real coefficients? But what about R, with the angled bracket? That I have no idea. XiXj = XjXi is perhaps an equivalence relation and the right hand side is the induced quotient set?
Edit: never mind, the video explained it: it's called a free algebra and l looked up the term on Wikipedia, sorry my background in abstract algebra is rather limited.
Ok, let me try to explain the notation:
- The coefficients are indeed the real numbers, as indicated by the R in front.
- Whatever is in the brackets (here X1, X2 and X3) are your variables.
- Square brackets [...] means that the variables commute.
- Angle brackets has no assumptions on commutation of the variables, e.g. X1X2 is not equal to X2X1.
I hope that makes some sense!🙂
@@VisualMath Crystal clear, thanks for explaining!
@@mingmiao364 Welcome 😀
Amazing!!
I agree, its fun.
I was taught the wedge product
measured the area of the parallelogram
made by the two vectors
and from that you can build
x wedge x is zero
as it has no area
and from considering (x + y) wedge (x + y)
which equals zero too
you end up with the anticommuting rule.
So I am not sure where is the best way to start
as they all circle the same fundamental qualities
Historically speaking you are correct. The geometric interpretation was the reason why people start looking at wedges.
I however personally feel that one can explain them in a more "modern way" as polynomials with signs, and the geometric interpretation comes then as an application.
Some people will argue that one can go one step further, "even more modern", and explain them via universal properties.
On reasonable spaces these notions all agree (which is remarkable), and each perspective has its advantages.
I just like polynomials ;-)
@@VisualMath
Polynomials are a fine way of introducing the exterior algebra
I suspect it was because
I encountered exterior algebra via the sciences
and a while ago
that they chose the "area approach"
It is remarkable that you get to the same system
from radically different starting points
It is indeed remarkable.
It is somehow a general, and surprising, "fact" that nice mathematics corresponds to nice XYZ (geometry in our example) and vice versa. That is of course not always the case, but close enough ;-)
frickin awesome
"Thank you, glad that you liked it!" (if you mean the video), or "Yes I agree!" in case you mean the exterior algebra ;-)
@@VisualMath definitely both
Trying to learn Rotors because Quaternions are too complicated for me. My journey has lead me here, and the topic seems like fun.
Not sure how these topics fit together, but awesome: welcome!
@@VisualMath So far as I've learned exterior algebra is the foundation for understanding bivectors
@@christelleaugustin1695 Hmm, I am not quite sure what you mean. What do you mean by bivectors?
@@VisualMath He was learning clifford algebra (probably under the heading 'geometric algebra'). bivectors are 2-vectors and his interest is in their use for representing arbitrary planes of rotation (hence rotors) through the generalized euler formula exp(-plane*angle/2) *(multivector being rotated)*exp(plane*angle/2).
@@TheRevAlokSingh Oh, thanks! I never heard the name "bivectors" for the corresponding element in the exterior algebra, but Dr google just convinced me that this is completely my fault.
I still can't think with planes instead of vectors. It's so confusing. Thank you for the video, anways.
Maybe the following helps. Take your hand as in this picture
en.wikipedia.org/wiki/Right-hand_rule#/media/File:Right_hand_rule_cross_product.svg
If you know the vector for your thumb and you want to have orthogonal vectors, then there is no unique solution. But you can rather turn your hand and get a whole plane, namely the one spanned by index and middle finger.
do you think you are a bumbling blue, ragnificent red or bumbling brown kind of guy?
I am neither of these, I am just red ;-)
@@VisualMath do you mean ragnificent red?
@@axog9776 No, I am just red and nothing else ;-)
@@VisualMath That is not acceptable. I hereby decree you as ragnificent red.
@@axog9776 Ah, its too late, I lost...