Matrix Groups (Abstract Algebra)
ฝัง
- เผยแพร่เมื่อ 13 เม.ย. 2014
- Matrices are a great example of infinite, nonabelian groups. Here we introduce matrix groups with an emphasis on the general linear group and special linear group. The general linear group is written as GLn(F), where F is the field used for the matrix elements. The most common examples are GLn(R) and GLn(C). Similarly, the special linear group is written as SLn.
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
www.jmilne.org/math/CourseNote...
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14 videos in a row.. I shall continue..
Never expected to become a enthusiastic of groups :-)
oh dear
The explanation was perfect. Thank you
Absolutely distinguish from others.
It's preserving joy of mathematics.
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Dear Google, please adjust TH-cam algorithm to favor this amazingly great channel and monetize it lavishly. Thank you.
From your lips to the GOOGLEGODS ears!! 💜🦉
Thank you ma'am. This is wonderful!
Thank you so much for these videos. You're able to break it down into nice and orderly chunks which is accessible even to laymen. This is a good refresher course for higher mathematics. Please make more!
Thank You so much!!!!
Thank you so much.
It's a new day where I am, and I'm already starting another binge-watch. I can't tell you how amazing your videos are, and how much I wish that I had come across them years ago...
Great explanation
your way of explaining is too good
can you add some videos on.cryptography and number theory??
Are you going to talk about representation theory? Or Lie groups and Lie algebras?
We'll definitely talk about representation theory. And eventually we'll talk about Lie groups, Abelian varieties, etc., but that's a ways off.
Is it possible to solve geometric problems in grouptheory using matrices? For example reflections + rotations of a triangle etc.? Or how do you rigurously prove that you have got back to the identity of the triangle for example.
Kindly upload more examples related to cyclic group normal group cosets etc , as soon as possible
a vídeo of the isomorphism theorems... :D please
Thank you for these videos! I wish my professor was this coherent!
For a field F and integer n: if we denote GLn(F) is the group of invertible matrices under multiplication with entries in F and denote SLn(F) as the subset of GLn(F) where each element has a determinant 1: then SLn(F) is a normal subgroup of GLn(F)
Yes, and there's a very lovely proof of this fact using kernels of homomorphisms :)
@@MuffinsAPlenty I just went this way: |CAC^-1| = |C| |A| |C^-1| = |C| |A| |C|^-1 = |A| = 1 where A is any matrix from SLn and C is any matrix from GLn and therefore CAC^-1 belongs to SLn and therefore its closed under conjugation from GLn and therefore its a normal sub group.
How'd you go doing it using homomorphism?
@@Yougottacryforthis What you did is a totally reasonable way to do it!
You can set up the function f : GLn(F) → F* (where F* represents the nonzero elements of F under multiplication) given by f(A) = det(A). Using the fact that det(AB) = det(A)det(B), we see that f is a group homomorphism. Then ker f is the set of elements A of GLn(F) which get mapped to the identity of F*. In other words,
ker f = {A in GLn(F) : det(A) = 1} = SLn(F).
Since kernels are always normal subgroups, we get the result!
It's not simpler than your method, but I like it because it's a nice example of how homomorphisms can be used in creative ways for things like this.
Thanks
Please what is the name of the music u used in the video
The name plss or something similar
thank you madam...............
can we multiply matrix groups of different order? if so ,how is it made? by the way excellent videos!
You can multiply two matrices as long as the number of columns of the first matches the number of rows of the second. But the members of a matrix group (with matrix multiplication as the group operation) will always be square matrix with the same number of rows and columns, since they need to be invertible.
Yeah ... U r quite clear
Can you please tell me that if we are given a set of real valued function whose domain and range is real number then will composition of function be group or not please help
Generally, speaking _no._
While composition is an associative binary operation with an identity function f(x) = x, real-valued functions from R to R may not have inverse functions.
If you restrict to functions that are both one-to-one and onto, then this smaller class of functions will form a group under composition :)
graet...!! thanks dear for sharing videos.
Your teaching is excellent. You should have another course on group theory.
¡Gracias!
Thank you so much for your kind support!! 💜🦉
@@Socratica thanks to you. Quality material
Could you plies, for easy reference, add group notations to the description of videos talking about or mentioning particular groups like Dn, Sn, GLn ?
I am enjoining the abstract algebra videos, but I'm also constantly getting lost with notation which is not familiar enough to me.
Like just now am not sure in which earlier video did you describe Sn.
Cross-referenced video links would also be nice.
Socratica I'm sorry if I'm getting annoying.
Good idea! We'll do that right away. Thanks for the suggestion.
Socratica I'm just wondering will quotient groups appear in the curriculum?
Definitely! We're getting ready to make videos on homomorphisms, normal subgroups and quotient groups.
Matrices are also used in quantum mechanics.
Why is complex under addition only is infinite? What is an example to non Abel own finite? You gotta use more examples and not just terms
teachers are many ,masters are few
"There is no truth. There is only rabbits, ..." Werner Heisenberg.
"its perfectly okay "..
Matrices ARE NOT all of infinite order as you say at:53
What we said at 0:53, was that matrices provide an example of infinite, non-abelian groups. We do not assert that matrices are of infinite order, or that all infinite groups of matrices are non-abelian. (The diagonal matrices, for examples, provide an infinite groups of matrices that is commutative.)
@@Socratica apologies