This comment makes me happy. The main reason for making these videos is to show people just how amazing and beautiful and mind-boggling all those connections are. It's great to know that we manage to convey those ideas. Thanks for sharing!
The moment you showed Cayley tables as matrices I immediately understood the connection between quaternions and Pauli matrices, and why they are what they are Holy shit. Mind blown. Well done. You have just connected higher dimensional numbers, group theory, and linear algebra just like that. Kudos, mate
I really like the connection between (computer science) graphs and matrices. I recently learned about spectral graph theory (en.wikipedia.org/wiki/Spectral_graph_theory) and generalized eigen decomposition. It's beautiful that two things that are both so abstract and seemingly different like dots & lines and grids of numbers can have such a deep connection; it's so rewarding to be able to switch between the graph or matrix perspective to understand something since (in my opinion) graph-like relationships appear EVERYWHERE in the world--tree structures and chain-like monoids are more so just nice specialty cases.
Derivative as a matrix is my favorite linear algebra example. For awhile, i treated linear algebra and matrix algebra as separate entities. But *Linear Algebra Done Right* by Axler showed thr derivative example on a monomial basis and my mind was blown. Then I proceeded to read the rest of that book. 10/10, would recommend.
I'm just a programmer by trade, not a trained mathematician, and I could be wrong about this, but I'm pretty sure it's pronounced eye-DEM-po-tent. Idempotency is used in programming to describe any operation where multiple applications of the operation changes nothing from doing it once. So effectively the same definition. :)
According to Wikipedia (en.wikipedia.org/wiki/Idempotence) you can pronounce it either way. It comes from Latin "idem" which was pronounced as in the video.
@@koenvandamme9409 Perhaps I'm reading the guide incorrectly, but I believe it indicates that it can be pronounced eye-DEM-potent or i-DEM-potent. I don't think it says it is pronounced idem-PO-tent.
The a,b,c powers are obviously imaginary number systems. First one is dual numbers, where imaginary part is non-zero but squares to a 0 anyway. Second one regular imaginary part of complex number where i^2 = -1, and the last one is other kind of complex numbers where i to nth (8 in case of C) power make it to 1. It's rotational group of bigger size. So I guess there are matrices that could represent them?
That's a great point! I just had a look and yes, those number systems also have matrix representations. See en.wikipedia.org/wiki/Dual_number and en.wikipedia.org/wiki/Split-complex_number .
Matrix A could be like squishing one basis vector and permuting the rest? Think derivatives but more general, in fact any upper triangular matrix with a diagonal completely 0 should work
If you want to learn more about this trick, you could look up "Homogeneous coordinates" on Wikipedia. It is the basis for projective geometry, a very powerful framework because it allows many more transformations (such as translations) to be performed by matrices.
Also ALGEBRA is a Non PRECISE tool bcuz it uses Abstract unit equal or whole parts which WONT gv precision BUT GEOMETRY does by using Physical units Wit different denominator meanings !
NICE Video explanation ! But this guy ws NEVA goin to introduce GEOMETRIC ALGEBRA or Geometric calculus aka Non Newtonian! Goto BiVector & SUDGYLACMOE Channel!
We are planning a series about geometric algebra. You would know this if you watched our overview video: ruclips.net/video/6ywt_rhxJfY/видео.html Also, there's no need to shout.
When you showed the homéomorphisme between Cayley tables and permutations, it was a pure 😮🤯moment
This comment makes me happy. The main reason for making these videos is to show people just how amazing and beautiful and mind-boggling all those connections are. It's great to know that we manage to convey those ideas. Thanks for sharing!
The moment you showed Cayley tables as matrices I immediately understood the connection between quaternions and Pauli matrices, and why they are what they are
Holy shit. Mind blown. Well done. You have just connected higher dimensional numbers, group theory, and linear algebra just like that.
Kudos, mate
Always happy to blow people's minds. These high-level connections are exactly what the channel is about.
Thank you!
I really like the connection between (computer science) graphs and matrices. I recently learned about spectral graph theory (en.wikipedia.org/wiki/Spectral_graph_theory) and generalized eigen decomposition. It's beautiful that two things that are both so abstract and seemingly different like dots & lines and grids of numbers can have such a deep connection; it's so rewarding to be able to switch between the graph or matrix perspective to understand something since (in my opinion) graph-like relationships appear EVERYWHERE in the world--tree structures and chain-like monoids are more so just nice specialty cases.
Derivative as a matrix is my favorite linear algebra example. For awhile, i treated linear algebra and matrix algebra as separate entities. But *Linear Algebra Done Right* by Axler showed thr derivative example on a monomial basis and my mind was blown. Then I proceeded to read the rest of that book. 10/10, would recommend.
Thanks for the tip, I will check out the book by Axler.
I'm just a programmer by trade, not a trained mathematician, and I could be wrong about this, but I'm pretty sure it's pronounced eye-DEM-po-tent. Idempotency is used in programming to describe any operation where multiple applications of the operation changes nothing from doing it once. So effectively the same definition. :)
According to Wikipedia (en.wikipedia.org/wiki/Idempotence) you can pronounce it either way. It comes from Latin "idem" which was pronounced as in the video.
@@koenvandamme9409 Perhaps I'm reading the guide incorrectly, but I believe it indicates that it can be pronounced eye-DEM-potent or i-DEM-potent. I don't think it says it is pronounced idem-PO-tent.
The a,b,c powers are obviously imaginary number systems. First one is dual numbers, where imaginary part is non-zero but squares to a 0 anyway. Second one regular imaginary part of complex number where i^2 = -1, and the last one is other kind of complex numbers where i to nth (8 in case of C) power make it to 1. It's rotational group of bigger size.
So I guess there are matrices that could represent them?
That's a great point! I just had a look and yes, those number systems also have matrix representations. See en.wikipedia.org/wiki/Dual_number and en.wikipedia.org/wiki/Split-complex_number .
@@AllAnglesMath I get the names from there, lol.
Matrix A could be like squishing one basis vector and permuting the rest? Think derivatives but more general, in fact any upper triangular matrix with a diagonal completely 0 should work
the 'fake' translation matrices were really interesting
If you want to learn more about this trick, you could look up "Homogeneous coordinates" on Wikipedia. It is the basis for projective geometry, a very powerful framework because it allows many more transformations (such as translations) to be performed by matrices.
Also ALGEBRA is a Non PRECISE tool bcuz it uses Abstract unit equal or whole parts which WONT gv precision BUT GEOMETRY does by using Physical units Wit different denominator meanings !
NICE Video explanation ! But this guy ws NEVA goin to introduce GEOMETRIC ALGEBRA or Geometric calculus aka Non Newtonian! Goto BiVector & SUDGYLACMOE Channel!
We are planning a series about geometric algebra. You would know this if you watched our overview video: ruclips.net/video/6ywt_rhxJfY/видео.html
Also, there's no need to shout.