Talk:General linear group

Application to polynomials

If p is a polynomial in F[x] of degree n, then we can consider p as an element of the vector space with basis elements (1, x, x², ..., xⁿ).

Late now... I'll finish this up tomorow...

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Here's an example of a general issue with the mathematics pages (and even any of the more esoteric topics) of Wikipedia which I'm struggling with.

By removing the section on "Motivations", Axel has condensed and clarified the article; which makes it easier to read - if one knows what's going on already in the topic.

But if someone is not quite sure what GL(n,F) is, or why it exists, (which is the reason that I presume they sought out this article) I find a sentence such as:

If the dimension of V is n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Once a basis has been chosen, every automorphism of V can be represented as an invertible n by n matrix, which establishes the isomorphism.

may not satisfy them; what is an automorphism of V? What is the relationship between V and n and F? What does it mean for an isomorphism to be canonical in this context? Exactly how do you apply an n by n matrix as an automorphism of V? How do you multiply a vector space by a matrix??? WILL THIS BE ON THE TEST!!????

Let me add that the above quoted statement is absolutely correct, and that I find it a clear explanation of the relationships I alluded to in the motivation section. (Also, there was an error in the original motivation (automorphisms of V should satisfy T(c'v)=c'T(v), not U(c)T(v)).

I already know the answers to all the above questions (and no, it will not be on the test :) ). My question is - does it serve the expected reader? Who is the expected reader, and how do we best serve them?

These thoughts came to mind in discussion with a friend of mine who is getting his masters in mathematics, and with whom I've been coaching regarding group theory in general, and linear groups as automorphisms of roots of polynomials in particular. While he might understand some of the mechanics of these issues, I find that where he needs the most assistance is in the acquisition of "gut" feelings or instincts about the role and use of some of these structures, and I've been writing (or at least attempting to write :) ) with his point of view in mind.

So I find myself somewhere between writing a textbook explanation of concepts on the one hand (too much info), and an encyclopedia/dictionary approach (assumes too little background for a naive user).

With any of the articles I write in this area, I am trying to satisfy all readers - the mathematically sophisticated, as well as those whoe are just starting the topic under consideration. So, for example, in an article regarding PSL(n,F) (affine transformations), I could see using language like the above; since once you're looking at PSL, you've got to be already pretty well versed in these issues; but GL is just general enough that I can assume knowledge of the terms automorphism, vector space, and so on; but not neccessarily a knowledge of how it all ties together.

I'll address this style question mainly to you, Axel; as I seem to be bumping into you editing away in the mathematics domain; but I'm curious as regards the philosophy overall of wikipedians. Chas zzz brown 20:57 Oct 20, 2002 (UTC)

Sorry, I was a bit rude in cutting the motivation section. There were a couple of things wrong with it; the given formula for T was not the general form of an automorphism and the field automorphism U was not really needed, especially in an introductory motivation paragraph. Then, the next section seemed to repeat that elements of GL are automorphisms, but talked about "automorphic space induced by M" which I couldn't make sense of. That the rows and columns of an invertible matrix form bases of Fⁿ is certainly useful information, but should probably either go into matrix or basis (linear algebra) where more people will see it.

Generally speaking, I want the articles to give useful information to amateurs and experts alike. Amateurs will usually only be able to understand the first couple of sentences (which should give the proper "gut feeling" and intuition), before they run into terms they don't understand and have to click on a link. That's just fine.

I agree that the quoted paragraph above about the non-canonical isomorphisms is not clear. This should be expanded, with an explicit construction of the isomorphism. On the other hand, people who know what a finite-dimensional vector space is usually also know how to represent linear transformations as matrices. AxelBoldt 22:29 Oct 20, 2002 (UTC)

Thanks for responding; no offence taken at the edit. I agree there were many problems with the section you cut (read: wrongness!), and had been thinking about how to correct them when I woke up; then I saw the whole section was gone. I just wanted to get some feedback on how to fix it from the point of view of usefulness; i.e., whether it was deleted for lack of correctness or from inappropriate context (or both).

As regards the style question, I suppose I look at mathworld.com as a sort of minimal standard; it leans more to the dictionary end of things. I'd like to see Wikipedia stand somewhat closer to the explanatory end than that; not that it should become a collection of proofs, but that it should contain plenty of intuitions regarding why certain structures and concepts exist.

I happened to be looking at the EPR paradox page today and saw the recurring "expert article" style problem; although understanding bra-ket notation implies an ability to understand EPR, the converse isn't the case; I have only a vague notion of the mathematics on that page, but I think I understand why E, P, and R were so upset. (A better explanation of why faster than light communication would be considered paradoxical would help there, but that's a different thread...).

It's hard to recall what some of this stuff looked like before I got my head around it; and that makes striking the right tone a bit challenging (at least for me :) . Chas zzz brown 23:07 Oct 20, 2002 (UTC)

I agree that we should have more explanations than mathworld (and less mistakes), and I also don't have anything against the occasional proof, as long as they are clearly marked and can be skipped. AxelBoldt 16:17 Oct 21, 2002 (UTC)

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We should have a few words about the projective groups, no? Someone care to take this on? Revolver

Rings

Can't this be defined for matrices over commutative rings just as well as fields? Mac Lane's book has that usage... A5 11:44, 3 March 2006 (UTC)

You can take the unit group of a matrix ring over any ring. What do you do next, though? We don't strive for ultimate generality. Charles Matthews 12:01, 3 March 2006 (UTC)

I don't understand what you're getting at. What part of the GL definition depends on having a multiplicative inverse? As for "we don't strive for ultimate generality" - for instance, people talk about SL2(Z) a lot, are you suggesting Wikipedia just ignore such objects because Z is not a field, or ...? A5 14:24, 3 March 2006 (UTC)

Even Serge Lang's "Algebra" defines it over a ring. I don't know where Wikipedia's definition came from, but it is non-standard and I think we should change it to be standard. A5 14:45, 3 March 2006 (UTC)

What part of the GL definition depends on having a multiplicative inverse? It doesn't. On the other hand, if you want GLⁿ(R) for R a general commutative ring, you might as well say it is a group scheme. I'm sure that was in the back of Mac Lane's mind. Since it works without the commutative law, are you not going to allow us to stop there? I'm sure you're wrong about the non-standard definition, though. General linear group over a field is a baic, useful definition. Charles Matthews 16:06, 3 March 2006 (UTC)

General linear group over a field is a basic, useful definition. Yes, and it is a special case of the correct definition, over a ring. I've explained what the costs of keeping the current definition are. The ring definition is used quite frequently - I've not seen any other - and it is used in many important special cases like SL2Z, and people will be confused if they look these up in Wikipedia and find that the Wikipedia definition uses fields instead. Please tell me why you object to using rings in Wikipedia's definition, or I'll go ahead and change it. What is the cost. A5 01:46, 4 March 2006 (UTC)

The definition over a field should be kept as the primary one. However, I think it would be appropriate to have a section (called Generalizations or similiar) on the ring-theoretic definition. -- Fropuff 07:15, 4 March 2006 (UTC)

Well, I certainly object, in all cases, to generality that makes it harder for the general reader to see what's going on. For GL of a ring there are numerous things that don't work. For example, PGL isn't so clear. So I agree with Fropuff. We can certainly speak somewhere about the GL that denotes a functor from rings to groups; but this is not appropriate for the entry definition. Charles Matthews 13:05, 4 March 2006 (UTC)

How about we say "GL_n(R) is the group of invertible n by n matrices over a commutative ring R. In most cases R is a field such as R or C." That way, users who don't know what a ring is will still be in familiar territory. Additionally, we will not change the PGL entry to make any reference to rings, to keep Charles happy. Is that OK? A5 21:47, 4 March 2006 (UTC)

It's still the wrong way to introduce the topic. We do have style guides for lead sections; we do have conventions about using 'concentric' style, where you tell the reader the short version, then the expanded version, then the still fuller version. That may not be the way graduate-level mathematics texts are written. But there is no need at all for Wikipedia to imitate such books. It is doing something different. Charles Matthews 21:58, 4 March 2006 (UTC)

That is starting with the short version. Most uses are over R and C, I think you'll agree.

No, I don't agree. The case of a finite field is fundamental, to finite group theory. And I still think you are confused about what we are doing here. We are not followers of Bourbaki, this is not a classroom, this is not a textbook treatment, and we are constantly asked not to make the abstract gradient steeper than it absolutely has to be. Charles Matthews 15:59, 5 March 2006 (UTC)

Additionally, it keeps things simple by not giving the definition in terms of a mathematical object with more operations than are necessary (i.e. a field). Should we say, under the entry for group, that a group is something with an associative, commutative, invertible, binary operation, only to mention as an aside half-way through the article that the operation need not be commutative? No, it makes it harder for people to learn the correct definition, and it forces people who just wanted a quick summary to read the entire article. The same considerations apply here. This is crazy. Besides, I don't see why you think that people who know what a field is, won't also know what a ring is. A5 22:08, 4 March 2006 (UTC)

Gotta admit I find it rather shocking that this article doesn't mention rings. It definitely should, somewhere. That being said, I think the starting posture of the article should be about GL(field), especially over R or C, because that's the most accessible to the widest audience. Dmharvey 22:52, 4 March 2006 (UTC)

I think the starting posture of the article should be about GL(field) OK, sure. But my feeling is that GL(field) should be presented as an early example, not as the canonical definition. If it is presented as a definition, it should at least be preceeded by something that says "For a more general definition in terms of rings, see ... below". However, I think users will find it a bit silly when they scroll down to the "more general definition" and the only difference they notice is that the word "field" has been replaced by the word "ring". It's overkill. A5 23:13, 4 March 2006 (UTC)

I don't think it's overkill. If you prefer, we don't even need to give a general definition that happens to be the same sentence with "field" replaced by "ring"; you just say "An important generalisation is where we replace "fields" by "rings"." The point of the latter section is not to give a definition; the point should be to explore some important examples, for example R = Z, R = a p-adic ring, R = Z/n (ha you can tell I do number theory!), point out that forming GL is functorial, discuss the relation to lattices and who knows what else. The principle I work by in these cases is that: (1) someone who doesn't know much doesn't get scared away, and (2) someone who knows more is experienced enough to know that the good stuff is a few pages down. Maybe you want to start a new article, like General linear group (ring theory), and think of this one as General linear group (linear algebra)? Dmharvey 23:29, 4 March 2006 (UTC)

I'd start with not mentioning either rings or fields, but just by saying "the general linear group of degree n, written as GL(n), is the group of n×n invertible matrices" (perhaps don't even say that it's a group). Then make another section with a precise definition, where you mention that the fields of entries needs to be specified, and that it can also be a ring. -- Jitse Niesen (talk) 00:18, 5 March 2006 (UTC)

Jitse: If you had said "the rings of entries need to be specified" then I might agree. Dmharvey: I think it's much more clear to say "an important special case is fields" than "an important generalization is rings". If you say the latter then people will wonder what else the generalization consists of, that it was too difficult to introduce at the beginning. You'll have to add some explanation like "The generalization is actually trivial; we could have been using the word 'ring' all this time, but we didn't want to intimidate you." just to make it clear what's going on...

Also, since I've never gotten an answer: (a) why are rings argued to be so much more "intimidating" than fields - and especially, why do we think this is the case for people who have some reason to view this article; (b) why stop at fields, why not give our objects some other irrelevant properties like uncountability or completeness, just in case some readers are really really timid? It's not like someone will have heard of finite fields if they haven't heard of rings. A5 02:38, 5 March 2006 (UTC)

For what it's worth, I think I like Jitse's solution, with A5's caveats. Trying not to scare someone by avoiding the mention of "rings" in an article like this just seems silly. And I really hate the idea of lying to someone (telling them it must be field) rather than not telling them the whole story (that fields are 99+% of the usage). wnoise 03:03, 5 March 2006 (UTC)

Linear algebra is done over fields because it is easier, more important, and better understood. Show me an undergraduate linear algebra class that defines modules first then gives vector spaces as a special case. It would be perfectly logical. It has never been established Wikipedia practice to start an article with the most general definition. But by all means, lets have a section on the general linear group over rings. Those who understand modules vs. vector spaces, and rings vs. fields aren't going to be confused by this layout. -- Fropuff 03:09, 5 March 2006 (UTC)

Show me an undergraduate linear algebra class that defines modules first then gives vector spaces as a special case. It's not really necessary to define modules in a class about vector spaces, that's why they are not defined. But when vector spaces and modules are defined in the same class, in my experience modules always come first. A5 03:29, 5 March 2006 (UTC)

Let me expand on my previous point. We like to do fields first because of the connection with vector spaces. GL(n,K) can be identified with the set of all invertible linear transformations of a n-dimensional vector space over K. What about GL(n,R)? It is the set of all invertible linear transformations of a rank n free (right?) R-module. Far more people are going to scratch their heads over this statement as compared to the vector space version. -- Fropuff 03:17, 5 March 2006 (UTC)

For a ring R with identity, GL(n, R) is the group of invertible n by n matrices over R. This is the definition in Lang's Algebra, Mac Lane's "Category Theory...", and Mathworld. A5 03:29, 5 March 2006 (UTC)

(via several edit conflicts :-) Concerning (a). Yes, rings are "more intimidating" than fields in this context. Just because they take less axioms to define does not make them easier to understand. The group GL2(Z) is *much* harder to understand than the group GL2(R). For much the same reason that it's easier to learn about varieties over a field than about general schemes; for the same reason that the theory of eigenvalues/vectors is easier over an algebraically closed field than is the theory of decomposition of modules over an arbitrary ring; for the same reason that this article currently restricts itself to finite-dimensional vector spaces; for the same reason that the derivative article only considers the high-school version of the derivative and leaves more general ideas to other articles.

Let me try to be more specific. I think at least the first part of this article should be accessible to a first-year undergraduate student. Such a student almost certainly doesn't even know what a ring is, let alone a field. They may or may not have seen complex numbers before. What they know is that "numbers" can be added, subtracted, multiplied and divided (that's right, divided). Their knowledge of linear algebra is limited to solving two-by-two systems of linear equations, and it has never occurred to them that the division you need to perform during that calculation is anything special, because for them 7/4 means exactly the same thing as 1.75. Perhaps they got led to this article because they've just started learning about matrices and such things, and it's already a giant leap to collect together all of the invertible matrices into one object and give it a name. For such a person, discussing general linear groups over an arbitrary ring will in actual fact cause them to stop reading and go elsewhere.

Anyway, the question is, apart from a modification to the definition, what else do you want to add to the article? Perhaps you could start a draft on a subpage, listing some of the facts you think should be included, and then when we have a better idea of where the article could be going, we can do a merge. Dmharvey 03:27, 5 March 2006 (UTC)

I'm not willing to do that. I've wasted far too much time on this already. I think that your priorities are very misplaced. There are probably 2 people in the entire world who (1) know what a field is and what a group is but not what a ring is; and (2) are interested in learning about GL; and (3) are not willing to read through a sentence containing the mysterious word "ring" to get to the following sentence containing the more familiar "field" and R and C in order to do so. And when I add, who (4) are worth obfuscating the article for; it drops to 0 people. A5 03:40, 5 March 2006 (UTC)

Well I've tried your suggestion. How does it look to you? Dmharvey 04:01, 5 March 2006 (UTC)

Dmharvey: It's OK with me, actually. It's very pedantic, but one doesn't have to read far for the general definition, which was my complaint about the earlier proposals. A5 15:09, 5 March 2006 (UTC)

You don't need to know about rings and modules (or fields and vector spaces) to play around with SL₂(Z) and lattices.--gwaihir 14:00, 5 March 2006 (UTC)

Current introduction looks good to me. Start with the easiest to explain example, then give the full generalisation, highlighting the importance of the Real and Complex cases. There could be a need for a section on the General linear group over a ring. There seems to be enough material in this discussion for this.

Also not sure about ordering of sections, seems like Special linear group and Other subgroups naturally follow on from the As a Lie group section. Maybe its better to move Over finite fields below these (are there interesting examples of these subgroups over finite fields or rings?). --Salix alba (talk) 13:29, 6 March 2006 (UTC)

FWIW, I agree with majority opinion here. When I was an undergrad, I had no no clue what a field or a ring was, and yet I was required to know a lot about GL_n and its subgroups. Like almost all physics students, I had a pretty good idea of what Lie groups were long before I'd heard of a module, and long before I stopped being scared of rings. One should assume that most readers will have the same general experience. linas 15:22, 6 March 2006 (UTC)

How about an analogy: many people have poor eyesight, and the default browser font size is too small for them. But these people can always make the font bigger. Sure, some people will be too lazy to do this, or maybe they won't know how - but although we shouldn't make it purposefully difficult for these people, their potential inability or unwillingness to configure their browsers is not a reason for Wikipedia to have a huge font by default.

Similarly, many people won't know what a ring is, and they'll be confused if a definition mentions it. To be sure, we shouldn't go out of our way to make things difficult for these people - we can say that a field is a special case of a ring, and that R is a special case of a field (and we have done so, and I'm happy with the result). Yet, as pointed out, this will not be enough for some users - perhaps they'll have such poor reading comprehension or poor motivation, that the first new word they encounter will make them give up, or lose interest.

But that's a personal issue which such users can and should deal with themselves - as long as they haven't addressed it, anything we do to try to work around it will only be marginally helpful to them. It certainly isn't a reason for Wikipedia to water itself down - and this is where my position differs from that of Charles Matthews, Fropuff, Dmharvey, etc. Aside from the fact that it would make Wikipedia useless as a reference work for experts, oversimplification also leads to misunderstandings and confusion among people such as myself who are still learning the field. Just as with the font size analogy, a policy of oversimplification would insult the vast majority of users, and an even larger fraction of contributors, for an extremely small gain. A5 15:16, 7 March 2006 (UTC)

We are not trying to "water things down". We're just trying to put things in an order which will be useful to audiences with a variety of backgrounds. Have you considered that the particular level of sophistication that you expect of the introduction to this article might be related to your own level of mathematical experience? By the way, if you don't mind me asking, what kind of mathematical background do you have? Dmharvey 15:44, 7 March 2006 (UTC)

Dmharvey: I don't think that in the above paragraph you are really confronting the points of the argument that I made. I feel that the things you say have been said and responded to already. Obviously I don't think that "watering down" is your primary motive. A5 02:03, 9 March 2006 (UTC)

In that case I don't think I understand your argument. Are you suggesting that the opening definition should always be the most general possible? Dmharvey 02:27, 9 March 2006 (UTC)

Yes, within reason. Note that, way at the top of this discussion, people such as Charles Matthews have argued that rings shouldn't be mentioned at all in this article; this I strongly disagree with - but I also disagreed with a suggestion that they go into a separate section. If there is a "standard" definition, then the introduction should give it; if there are multiple common definitions, then the introduction should mention them all. A few people seemed to suggest that giving general definitions in the introduction is a bad idea because some users are frightened or confused by the sight of things which they don't know yet - the argument seemed to be that we should help them out by omitting or secluding potentially frightening pieces of information. In my penultimate comment, I made an analogy between this group of users and another group of users, with the purpose of illustrating why "hiding" information for their benefit is the wrong policy in an encyclopedia. Clearly I am not opposed to making articles more accessible, but I think that this should be done by adding new information, not by obscuring or by oversimplifying. Also, you seem to be stuck on the idea that this is just about levels of education - it isn't. As I have tried to argue before, people who have good reading comprehension will have no trouble skipping over the mention of "ring", whether or not they know what a ring is, when it is pointed out to them that something they are familiar with is a type of ring. If we hide information, we do so not on behalf of those who are uneducated, but on behalf of those who are uneducated and have poor reading skills - a small population, who would be better assisted in their development by a personal instructor. Please let me know if I have misunderstood your position, as you seem to have taken my comment personally. A5 04:24, 9 March 2006 (UTC)

First, don't worry, none of this is taken personally. It's actually a very interesting discussion, and I think this is one of the deepest issues we face in working on wikipedia, which is why I'm devoting so much time to discussing it :-)

I still don't buy your argument. Why don't we start by saying "Given a vector space V over a field K, the general linear group GL(V) is the group of automorphisms of V", which frees us from the restriction of finite dimensionality (and incidentally from some non-canonical choice of basis)? How about: "Given a Banach space V over a field K, the general linear group GL(V) is the group of continuous automorphisms of V". Do we need to cover this case in the introduction too? Where do we stop? What is your basis for the (implied) claim that the definition in terms of matrices over a general ring (commutative? associative? unital?) is the "most common" or "standard"? Dmharvey 13:12, 9 March 2006 (UTC)

What is your basis for the (implied) claim that the definition in terms of matrices over a general ring (commutative? associative? unital?) is the "most common" or "standard"? First of all, it includes the definition over a field. The definition over a field doesn't actually make use of the field structure - it is no different from the definition over a ring - so it seems silly to say "field" instead of "ring". That's a very small and natural change. Furthermore, there are many uses of GL(n,R), even on Wikipedia. Finally, the ring definition was the only one I found when consulting several sources. All this has been said before. I don't know about these other ideas you have brought up, but they seem like straw men. A5 23:25, 9 March 2006 (UTC)

Serre's Linear Representations of Finite Groups begins by defining GL(V), for an arbitrary vector space V over C, as the group of isomorphisms from V to itself (as I suggested above, but for a general field). Later he specialises to the case of a finite-dimensional V, and points out that in this case you can define the group in terms of matrices. Do you think perhaps we are misleading our readers by not including this non-finite-dimensional case in the introduction? Why is this a "straw man"? Dmharvey 00:38, 10 March 2006 (UTC)

There is a big difference between a "generalization" that involves removing an unnecessary condition, and a generalization that involves a fundamental change of notation/perspective. A lot of the arguments that I made for GL_n(F) vs. GL_n(R) don't apply to GL_n(F) vs. GL(V), which is why I suggested it was a straw man. But since GL(V) is discussed immediately following the introduction, I don't think it is necessary to mention it; otherwise I think it would be. Alternatively, the first section could be moved into the introduction, since it is short and defines a notation. As for continuous transformations in a Banach space, shouldn't it be enough to mention in the Banach space article that automorphisms are assumed to be continuous? A5 19:08, 10 March 2006 (UTC)

Hang on a second. When going from GL(n, F) to GL(n, R), we're basically dropping the field axiom that says that elements have to have multiplicative inverses. When going from GL(n, F) to GL(V), we're dropping the assumption that the vector space has to have finitely many dimensions. Can you explain why, as you seem to be suggesting, that dropping the first assumption is "removing an uneccessary condition", whereas dropping the second assumption "involves a fundamental change of notation/perspective"? (If it would make you feel more comfortable, let's write GL(V) as GL(n, F) where n is allowed to be an infinite cardinal; or for emphasis, perhaps I should say "a cardinal without the unnecessary restriction of finiteness".) Dmharvey 20:47, 10 March 2006 (UTC)

I don't know where we're going with this since GL(V) is already mentioned in the article and I said that I think it's an important definition. But from the article: If the dimension of V is n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Once a basis has been chosen, every automorphism of V can be represented as an invertible n by n matrix, which establishes the isomorphism. I think this "the isomorphism is not canonical" makes clear how GL(V) is a different viewpoint/notation. Also, I don't think that it is as simple as defining GL(V) as GL(n,F) for some possibly infinite cardinal n. I'm not an expert, but I think bases are usually defined to be things such that every element of the space can be expressed as a finite linear combination of the basis elements. So does that mean that your infinite matrices can only have finitely many non-zero elements in each row? What about each column? This isn't covered in the matrix article. And how do I go about choosing a canonical basis for, e.g., the space of sequences N->R (which can have infinitely many non-zero entries)? It raises a lot of issues. Maybe you can unify the two concepts but I hope you agree that doing so isn't as trivial as saying "hey this multiplicative inverse condition isn't really necessary". A5 17:53, 12 March 2006 (UTC)

The article intro looks more or less okay at the moment. I just wanted to add a comment in support of A5. Defining GL_n(R) for an arbitrary (associative and unital) ring is perfectly natural and sensible and hardly more difficult or less accessible in any serious way than defining it for a field. It is probably safe to say that students interested in learning about GL_n who have no idea what a ring is are due to spend ten minutes looking at the definition, seeing as how they've entered 'the algebra zone'. Of course, it's still good to introduce the concept via a well-known and important special case, as the article does. Incidentally, for an example of an undergraduate text that does define modules first and then introduces vector spaces as a special case (all in a quite sensible way), see Elements of Abstract and Linear Algebra.  — merge 12:17, 8 March 2006 (UTC)

Do not confuse a pedagogical technique with an encylopeadia. If you have a class-full of undergrad math majors who signed up for a first course on algebra, that would be a great way to do it. If you have a someone who is *not* a math major, say, for example, an engineering student, who doesn't know what GL_n is, and needed to look it up on wikipedia, is self-motivated enough to go look it up, I think its safe to assume that they will have a rather poor math background. Don't screw over somebody who is motivated and wants to learn, by throwing "rings" at them early on. Learning what a ring is is a *lot* harder than changing the size of your fonts on the browser! linas 03:24, 9 March 2006 (UTC)

Linas: Please read my above reply to Dmharvey, just posted. It may make what I am saying clearer; you seem to be misinterpreting the analogy. A5 04:28, 9 March 2006 (UTC)

I am following your conversation with dmharvey. I think dmharvey is right. There are many strong advantages to a definition having GL(V) with V either a hilbert space or a banach space. Its easier to learn what a hilbert space is than to learn what a ring is; most students will know hilbert spaces pretty well long before before they hear of a ring. However, even that is too high a hurdle.

Look, why don't you read the conversation below, about determinants. I think it makes clear the pitfalls of starting with a ring-based definition. So, does the naive reader have to keep in mind that some of the results, but not all, hold only for commutative rings? Maybe some of the results and facts hold only for integral domains. Which ones? Does the reader have to learn what a domain is before they can grok what GL(R^n) is? How much of ring theory is needed? Does one need to know about ideals, or prime ideals? If not, why not? linas 01:15, 10 March 2006 (UTC)

I wasn't the one who suggested that we generalize to non-commutative rings, I merely asked what the correct thing to do should be. I'm not even sure what the "standard" definition for matrix multiplication for matrices over non-commutative rings is. The person who extended the generalization to non-commutative rings was Charles Matthews. I think we should try to do it if it makes sense, but not if it doesn't. The definition of determinant is the same over a commutative ring and a field. So, does the naive reader have to keep in mind that some of the results, but not all, hold only for commutative rings? The naive reader should be expected to have the ability / common sense / motivation to ignore things which we indicate he can ignore, and not get derailed. If a result doesn't hold for rings, we should just say so. My suggestion for the current state of the article might be to go back to "commutative ring" in the intro and have a reference elsewhere (to a subsection, or another article) for non-commutative rings. Obviously generalizations have to be evaluated based on the complexity they add to the presentation. If they are in common use, then they will have to be included in Wikipedia at some point, the question is whether they overlap enough with what is in this article to be made a part of it, or whether they should be treated separately. But in no case should "it might scare the uneducated" be a reason to avoid mentioning the existence of something. A5 19:08, 10 March 2006 (UTC)

Umm... I'm just a senior undergraduate, but I think the layout of the vector spaces article is helpful for me. It talks about generalizations such as modules in a subsection. Would it be such a bad idea for this article to discuss what happens in generalizations of GL in a subsection also? Anon 05:44, 3 September 2009 (UTC)

What kind of rings?

We say "commutative ring", Mac Lane discusses GL_n 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)

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)

Above. Like I said before, it would be the unit group in the matrix ring. Charles Matthews 15:50, 7 March 2006 (UTC)

Maybe it should be mentioned that the definition of SL_n cannot be carried over in a similar way.--gwaihir 16:13, 7 March 2006 (UTC)

Can one even make sense of a determinant over a non-commutative ring? Dmharvey 16:29, 7 March 2006 (UTC)

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)

But the same construction for C doesn't give SL_n(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)

While this is all very interesting, it is seeming a bit beyond the scope of this article, maybe we need a seperate article General linear group over a ring, or possible expand the stub Matrix ring? --Salix alba (talk) 15:00, 9 March 2006 (UTC)

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)

Warming to A5's position

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)

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)

Proof that SL(n,C) is simply connected?

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)

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 π_n(E) = π_n(F) x π_n(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)

As an aside, isn't SL(n, R) often also called SO(n,R)?

No. SO(n,R) is another group, the special orthogonal group. -- Jitse Niesen (talk) 09:02, 25 June 2006 (UTC)

Semidirect product structure

I've never edited a talk page before and have no idea where to put this. However, I noticed that someone claims GL_{n}(F) is the semidirect product of F with GL_{n}(F). While this is true in fields F where every element has a square root (also in R and probably many other fields, eg ordered fields where positive elements have square roots), I believe it isn't true in general for n \geq 2. In particular it isn't true for n = 2, F being the finite field with say 13 elements. In that case, GL_{n}(F) is (I think) the direct product along with a (nontrivial) group extension by Z_{2} (there should only be one of those possible). I'm sorry for not editing in the 'standard' way. I'm a long-time wikipedia user but have never been involved in article creation. My apologies as well if I in fact made an error here. —Preceding unsigned comment added by 171.64.38.45 (talk) 17:30, 24 October 2007 (UTC)

No problem. I added a heading to separate it from the previous discussion and removed a stray newline. I think the second time you mention GL, you mean SL, but I didn't want to change your text.

I can confirm that GL(2,13) is definitely a semidirect product of SL(2,13) and a subgroup D of GL(2,13) isomorphic to the group of units of the field with 13 elements. Specifically D is all diagonal matrices whose 1,1 entry is nonzero and whose 2,2 entry is 1. This group has 13-1=12 elements, and multiplication is component-wise, so isomorphic to the multiplicative group of the field. The intersection of D and SL(2,13) is all those diagonal matrices diag(d,1) whose determinant is 1, but the determinant of diag(d,1) is d*1=d, which is 1 if and only if d=1. In other words, the intersection of D and SL(2,13) is the identity subgroup. Furthermore, SL(2,13) is normal, being a kernel, and the order of GL(2,13) is the product of the order of D and the order of SL(2,13). This can be verified directly, or by using the determinant homomorphism. Here is a GAP session to show a computer verification, if you want to test similar examples, though the definition of D and the verification that GL = D |x SL is the same for any finite field. For an infinite field, you want to use the homomorphism idea more than the "just check the sizes" idea.

gap> G:=GL(2,13);

GL(2,13)

gap> S:=SL(2,13);

SL(2,13)

gap> D:=Group([[Z(13),0],[0,1]]*One(GF(13)));

Group([ [ [ Z(13), 0*Z(13) ], [ 0*Z(13), Z(13)^0 ] ] ])

gap> Intersection(D,S);

<group of 2x2 matrices of size 1 in characteristic 13>

gap> IsNormal(G,S);

true

gap> Size(D)*Size(S)=Size(G);

true

That's enough to verify the definition for a finite group. JackSchmidt 18:15, 24 October 2007 (UTC)

C^*

Since in almost every articles of research in mathematics the product group of C is denoted with exponent the star, I suggest to follow this notation, even if it should be less "naturally" than the notation with the times "x". Emc2fred83 (talk)

Preface contradicts first section

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)Santropedro

Topology of the general linear group

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)

A section on history and application of the general linear group (and special linear group)

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)

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)

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)

What about Linear group?

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)

Invertible in R is not always invertible in F

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)