Talk:General linear group

message: We say "commutative ring", Mac Lane discusses GLn as a functor on commutative rings, Lang defines GL in a chapter dealing with commutative rings; but Mathworld only says "ring with identity". Obviously an identity is necessary. Is commutativity necessary (for a sensible definition of matrix multiplication)? Do we care? I don't have enough background to say... A5 15:09, 5 March 2006 (UTC)
replies: text: I think Charles removed the "commutative" qualifier from the article. Where can I find a definition of invertible matrix for noncommutative rings? A5 15:19, 7 March 2006 (UTC)

text: Maybe it should be mentioned that the definition of SLn cannot be carried over in a similar way.--gwaihir 16:13, 7 March 2006 (UTC)
replies: text:  Can one even make sense of a determinant over a non-commutative ring? Dmharvey 16:29, 7 March 2006 (UTC)
replies: text:  Not really. To define the group SL(n,H) one embeds GL(n,H) in GL(4n,R) and uses the real determinant. -- Fropuff 17:42, 7 March 2006 (UTC)
replies: text:  But the same construction for C doesn't give SLn(C), so this is apparently not a generalization, but a different concept. (It is unclear to me how this should be done for a general ring, anyway.)--gwaihir 23:22, 7 March 2006 (UTC)

text: I wouldn't worry too much about it for now. So far we have zero content in this article pertaining to rings. If the material grows too large we can splice it off. I would very much like to see the stub at matrix ring expanded. Any takers? -- Fropuff 05:23, 10 March 2006 (UTC)
thread_title:  What kind of rings? 

message: What A5 is saying is making more and more sense. My argument appears to be going down in flames. I might go crazy with a rewrite. Let's see.... Dmharvey 19:36, 12 March 2006 (UTC)
replies: text: Well, I'll be darned... Thank you, I've never had an argument end that way before. :) In any case, I quite like the new version. A5 20:53, 15 March 2006 (UTC)
thread_title:  Warming to A5's position 

message: Anyone know a reference for this?
:This is essentially the same result as that the special unitary group is simply connected (which is mentioned in that article). That is because the two groups have the same homotopy type (which can be proved by Gram-Schmidt or the Iwasawa decomposition). By the homotopy long exact sequence this is the same as saying the unitary groups have fundamental group that is infinite cyclic. So we have all three results on the topology standing or falling together. I suspect that the special unitary case is easiest to prove from scratch, using induction and the action on complex projective spaces. Charles Matthews 18:03, 15 May 2006 (UTC)
replies: text: To show that SU(n) is simply connected, use the fiber bundle SU(n-1) -> SU(n) -> S^(2n-1) gotten by looking at where a fixed vector goes. Then use the fact that F -> E -> B implies πn(E) = πn(F) x πn(B), that π1(S^{2n-1}) = 1 for n > 1, and induction. The base case is SU(1), which is a point; then SU(2) = S3, etc. [Following Hatcher.] Tesseran 05:09, 31 May 2007 (UTC)

text: No. SO(n,R) is another group, the special orthogonal group. -- Jitse Niesen (talk) 09:02, 25 June 2006 (UTC)
thread_title:  Proof that SL(n,C) is simply connected? 

message: In the preface, the first part of the entire article, it says at the beggining
"In mathematics, the general linear group of degree n is the set of n×n invertible matrices"
And then in the first section called "General linear group of a vector space" it says this statement that obviously contradicts the former quoted, because matrix is different from linear transformation:
"If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V"
This is very confusing. I know the relationship between matrix and linear operators, but we need a better definition.Santropedro1 (talk) 18:34, 23 September 2015 (UTC)
thread_title:  Preface contradicts first section 

message: It would be great for this article to have a section on the topology of GL(n,R), the way the article on the special linear group does.  — Preceding unsigned comment added by 90.207.21.248 (talk) 22:58, 20 June 2019 (UTC)
thread_title:  Topology of the general linear group 

message: There is already a short remark on the history of the concept, stating that Galois defined the general linear group when investigating the Galois group of an equation. I think the article would benefit from extending this section (and maybe adding some remarks about the usefulness und applications of the concept). In particular, why is the general (special) linear group interesting? For what do we use it today? Why exactly was it introduced by Galois, why is it helpful when investigating the Galois group of an equation? Zaunlen (talk) 14:46, 10 November 2019 (UTC)
replies: text: Go ahead and add what you see fit! As a general remark I think it is important to stick to reliable sources when it comes to mathematical history. For example, I would like to see a reference for the assertaion that Galois defined the general linear group if this assertion appears in the article. Jakob.scholbach (talk) 11:04, 11 November 2019 (UTC)
replies: text:  I myself don't have the knowledge to write this section, it just was a suggestion that maybe someone else can execute. Zaunlen (talk) 19:23, 14 November 2019 (UTC)
thread_title:  A section on history and application of the general linear group (and special linear group) 

message: Should there really be two distinct articles, General linear group and Linear group? Both articles are about the general linear group and its subgroups.  
Should the other article be merged into this one? Sylvain Ribault (talk) 21:23, 1 December 2020 (UTC)
thread_title:  What about [[Linear group]]? 

message: mat
array([[0, 1, 1],
       [1, 0, 1],
       [1, 1, 0]])
numpy.linalg.inv(mat)
array([[-0.5,  0.5,  0.5],
       [ 0.5, -0.5,  0.5],
       [ 0.5,  0.5, -0.5]])
numpy.linalg.det(mat)
2.0
The article mentions GL(3,2) as an example. Its elements are the 168 invertible 3×3 matrices with elements in F2. It may not be obvious that whether or not a matrix is invertible depends on the field. The binary matrix shown on the right is invertible, but not part of the group. I suppose that one would describe it as "invertible in R" but "singular in F2". Maybe this should be clarified in the article. Watchduck (quack) 12:20, 30 August 2022 (UTC)
thread_title:  Invertible in R is not always invertible in F