Talk:General linear group
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I put this here for now:
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...
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)
We should have a few words about the projective groups, no? Someone care to take this on? Revolver