The General Linear Group, The Special Linear Group, The Group C^n with Componentwise Multiplication
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- เผยแพร่เมื่อ 31 ต.ค. 2018
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The General Linear Group, The Special Linear Group, The Group C^n with Componentwise Multiplication
This was great but according to my Algebra book and other sources, the General Linear Group is not limited to 2x2 matrices but literally nxn (so you have a GL group for each n). Might be a trivial distinction but as an student trying to piece together all this stuff it sounds important to me!
yeah it's for nxn matrices, the value of n is called the degree, so , so like GL_2(R) is the general linear group of degree 2, it contains all invertible 2x2 matrices with entries in R. Another example is GL_3(C), this is degree 3, and the entries are complex numbers from C, etc.
Super helpful! Thanks for taking the time to break it down bit by bit.
You are welcome!
I am wondering what can be the form of Matrix that belongs to special linear groups such that when multiplied with e (Standard basis for complex vector space) gives e ??
How about the set of complex n-tuples excluding those with any components with magnitude 0, under component-wise multiplication? That's a group? Trying to come up with examples of my own
I have just been exploring Lie Groups and Lie Algebras this week. I recognise some of those terms on boards.
wow nice
The Math Sorcerer I am only familiarising myself with the sight of maths inside Lie Theory and with mathematical terms. That done, I will retirn months or year later. . I am doing the same with other theories.
PS, I willl make a map of Lie Theory in Linoit, a open source website where you can create pinboards.
I'm wondering how we're able to use the identity matrix in the linear group since although its det = 1, its entries are real, not complex. Thanks for all your great content!
every real number is also a complex number 1 = 1 + 0*i
This is good content
Thank you! I will keep adding more:)
Hello, the example of general linear group in which book can I find it?
excellent
hey thank you!!!!
Good
Have you heard of the lie group SO(3)?
At 14:53, you refer to C^n as C to the nth power. I don't think you mean that, because the group is comprised of componentwise addition, not exponentiation. More generally, thank you very much for all of your math videos.
In a group the n^th composition of an element is often referred to as the n^th power of that element, so your exponentiation is the n^th composition in the multiplicative group, whereas I think here he means the n^th composition in the additive group.