I mean from every vector space you can create the set of all its linear non-singular endomorphism (GL(R,n) in our case) and since GL(R,n) itself is a vector space does this mean tha we can create an infinte chain of vector spaces created from the previous one?
#R^n gives birth to GL(R, n). Does this mean that GL(R,n) can give birth to GL(GL(R,n),n)?
languid nymph
spark jungleBOT
I mean from every vector space you can create the set of all its linear non-sing...
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languid nymph
And if yes, is it ever possible for such a chain to ever stop eventually? ( i.e. V ~ GL(V,n) in some way or idk)
languid nymph
wha
slate imp
did I accidentally speak spanish?
GL(V) is the group of invertible linear maps, implicitly V is a vector space
GL_n(R) isnt a vector space
so its general linear group isnt a thing
Like you just read the definitions and see it's nonsense
languid nymph
My bad, GL(R,n) is a vector space with + being matrix multiplication and * being matrix multiplication with a scalar
slate imp
no it isnt
languid nymph
It isnt?
languid nymph
Glaring problem?
slate imp
I was thinking of rings
but like... yeah, clearly that wont give a vector space structure
languid nymph
Ok new idea
We restrict GL to a subspace such that the matrices commute with each other
It can't be that much smaller than GL, can it?
slate imp
You trivially get the copy of R^x at the bare minimum
actually no, cause your scaling will always force 0 to be in the set
so * can never have an image contained in GL
languid nymph
sad
hardy fractal
It's a module over the ring of Gl_n(R) not exactly a trivial question
Replacing A with GL_2(A) for example is a cute way to proof a bunch of intresting things about Gl_n(A) for a ring A using whiteheads lemma
What you seem to be eventually reaching is called Gl(A) the infinite general linear group for a ring A it's not constructed exactly how you've written it but I'm assuming that's vaguely what you were thinking of
Gl(A) arises as a direct limit but you can talk about it purely in linear algebraic terms too just Google "infinite general linear groups"
Also yes Gl_n(Gln(R)) is well defined it's just a module not a vector space