Lie Groups #1 - Introduction + The General Linear group GL(n, R)
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- เผยแพร่เมื่อ 8 ต.ค. 2020
- This video will introduce the concept of a Lie group, realised as a continuous group that also has the structure of a smooth manifold (see my manifold lectures for more, I will assume you have some idea about what a manifold is in this video).
I will mostly be relying on examples rather than formal definitions to begin with, and this video covers the most general Lie group, the general linear group. I discuss how we realise this matrix group as a manifold (by constructing charts, effectively the matrix elements are the coordinates).
If you like my videos and want to consider supporting the channel I am now accepting donations in DOGE and BTC (other cryptos on request) See my channel description for the addresses!
I once again have a question and I'm sorry if it's a bit pedantic. But if the determinant of the matrix cannot be zero, that must mean there exist many combinations a, b, c and d (or more for higher n of course) that are excluded from this R^n² space. So then does this space sort of have "holes" in it? And is this a significant detail at all?
Great question and observation! Yes you are correct since we do have the restriction imposed by (eg for the 2x2 determinant) ab = cd.
I think I was too lenient in calling the matrix elements completely arbitrary in this case. We can talk about the set Mat(N) which would be the set of all possible NxN matrices, but this is not a group since not all matrices have an inverse. By imposing the condition the matrices be invertible, we do get a group, but then we do have to drop some of the possible matrices as you said. So yes these would be the 'holes', however, these would then be holes in the set Mat(N) rather than GL, (since GL is what is left of Mat(N) after the holes have been removed). Hope that helps!
Little pockets of infinite space to add more information if we can see through the zero...
didn't know that Joffrey Baratheon will teach me about Lie group
"proof by waving my hands." Like it! :) I do that a lot too.
Ure dog must be a genius mathematician by now. 😀 great explanation. 👏
I really appreciate the brevity and digestibility of these videos. As someone who just wants to learn abstract algebra concepts for practical reasons and not necessarily practice it in a formal academic setting.
way more clear than the other videos I tried, thanks for posting this
Thank you. It’s very helpful at times to dispense with formal definitions.
You can open Patreon where people will support you like crazy
Ive been looking for it
Great STUFF!!!
thanks for these nice lie group series! It is so helpful that instead of mathematical rigor, you're focusing on providing and intuition, totally love that! It would be helpful to see the similar set of lectures but just for group theory.
Thankyou! And yes I wanted to have a healthy balance of intuition vs rigor, as I find the majority of sources on these topics are incredibly dense and terminology laden, lacking any actual explanations or intuition! I have some older videos talking about algebraic structures, groups and vector spaces that might be useful! I also have some videos about the unitary groups and spinors that I still need to get round to editing!
講得不錯！
Hello, really great videos. Your intuition based way of teaching is really helpful (the manifold videos were especially good). Not sure if this is something you'd be interested in but Eric Weinstein (mathematics PHD working with Peter Thiel) is building a community to crowdsource information regarding mathematics, geometry and physics. His project is called The Portal. I think they would really welcome someone with your skillset. All the best.
Thank you.!
Obrigado.
Jack Gleeson left acting and now teaches Lie Groups on TH-cam. Great
"proof by handwaving" :D :D Hope my Prof. will be ok with that, too :D hahaha
It's my favourite proof method! 100% effective ;)
As an aside, I'm not sure if you were aware of how difficult your channel is to find via a cursory search of the channel name on youtube. The channel is extremely warm and inviting and it's a shame that youtube hides it so deeply within the algorithm.
Yes I have experienced this:( it's a real pain and I think it's because the name of my channel is all one word...I might split is and see if that helps, thanks for letting me know and I'm glad people are searching for my channel anyway!!
Loved the video, the wobbling is a bit distracting especially at when watch at 1.5 speed
My god it was driving me crazy.
Apart from continuous symmetries there are discrete symmetries. Is there a reason why you only discuss the continuous ones?
Nice video but could you please stand still?
what's the meaning of all those decorations? is it christmas special lmao? But what's that dog for!!!!!!!!! I mean seriously dogs and lights in a maths video!!!!
Good content but please stop dancing while you speak.