Linear Algebra and Group Theory

I need to know if I understand subspaces correctly so if anyone can point out any flaws in my thinking it would be well appreciated... If you have 2 vectors, each with 3 real numbers, the vectors exist in [math]R^3[/math]. Since there can only be 2 pivots at the most, the subspace that the vectors generate is in [math]R^2[/math], but not any higher. A question here: Is there any way that the subspace could exist in [math]R^3[/math]? If you have a set of vectors and you're looking to generate a basis for a subspace, the vectors must be linearly independent, that is, having the trivial solution [math] 0 [/math] as its only solution when the vectors are augmente…

Does anyone know if DVDs on how to teach yourself Algebra and Calculus really work? I'm looking into buying DVDs to teach myself calculus, even though I am on a budget. Do DVDs work?

Barnoulli's principle states that pressure of static fluid>pressure of moving fluid.So if water is made to rush at a greater pressure than that in the ocean,then the water-pipe will contract such that the water-supply will be stopped,resulting in bursting of the pipe.PLEASE COMMENT.

How can we say that , or can calulate a group is cyclic or not? Please need help

Has anyone worked on the P vs NP problem ?

So I'll be taking Linear Algebra come January, and I always like to be prepared or at least a little familiar with material I'm about to encounter. About linear algebra, I have no clue...I know it deals with matrices, at that's it, plus I'm a little rusty on those as I haven't worked any since high school. Can anyone give me a heads up on some of the general ideas of the course, what to look out for, or maybe even a recomended book?...ty

I've read somewhere that a unitary matrix U can be defined by the property: (1) U*=U^{-1} (* = hermitian conjugate) or by the fact that it preserves lengths of vectors: (2) <Ux,Ux>=<x,x> I have trouble seeing why they are equivalent. It's obvious to see that (1)=> (2): <Ux,Ux>=(Ux)*(Ux)=x*(U*U)x=x*x=<x,x> But not the other way around. I CAN prove it for real vector spaces, where U is an orthogonal matrix from the fact that <v,w>=<w,v>. Then I would do: <v+w,v+w>=<U(v+w),U(v+w)>=<Uv,Uv>+<Uw,Uw>+2<Uv,Uw>=<v,v>+<w,w>+2<Uv,U,w> and working out the left side gives <Uv,Uw&…

I'm having a little trouble with some of my homework here and need a little guidence. The problems are both proofs and I have a lot of trouble deciding what a conclusive proof is. I'm looking for a little hint on what might be a good first step or two in showing the following two things: 1) Let (v1...vn) be a spanning set for the vector space V and let v be any other vector in V. Show that v,v1,...,vn are linearly dependent. I'm just having trouble conclusively showing in this one that the coefficient on v must be some combination of the others. I know it's true through common sense but I can't prove it. 2) Let V be a vector space. Let v1, v2, v3 and v4 be vec…

This article was very appealing to me.Only thing is that one has to read it to completion(including the links provided in it) to appreciate it's true beauty. http://en.wikipedia.org/wiki/Srinivasa_Ramanujan

A very interesting problem I found...

Can someone please tell me what logistic regression is? I know nothing of regression also (assuming that something like that exists). I read wikipedia, but couldn't understand much. what exactly the following line means: "Logistic regression is a statistical regression model for binary dependent variables." Secondly,can someone please explaine each and every term of the equation given by wikipedia. Link is given below. http://en.wikipedia.org/wiki/Logistic_regression Thanks in advance

I'm curious about something, whenever we count in a certain base we have to be able to count upto that number, example in base ten we count from 1 to 9 then proceed to one zero, or 10. same as binary. now in that respect I can see why counting in base e or base infinite would be useless but what if we counted in powers of infinite? now i realise that this is impossible unless it is infinite to the power infinite and so forth. now this may be just a mathematical side thought but i was just curious if iwas staying in the realms of mathematical rule.

I have little background in number theory, groups, conditions, and stuff like that. So I got a book called Teach Yourself Mathematical Groups and this is one of the examples. Prove that a necessary and sufficient condition for a number N expressed in denary notation to be divisible by 3 is that the sum of the digits of N is divisible by 3. I see the proof in the book, but I can't get it. It shows what denary notation is, decimal notation written out like 1x10^4 + 2x10^3 ... Let N= a 10^n + b 10^(n-1) + ... + z (The book uses subscripts instead of different letters for the digits a, b, ..., z) The proof says: If 3 divides N, then 3 divides a 10^n + b 10^(n-1…

Can you imagine a group with infinitely many identity elements? A structure like this have discovered mathematician Algirdas Javtokas, he calls it a beta group. I have read his axioms in Beta Algebra book, but still it is hard to understand how such structure can “work”. Maybe you know something more about this?

the mathematician john baez has an online blog-like column with some entertaining math facts and often some surprisingly beautiful pictures here is the latest "Week" #228 (he doesnt write them every week, only now and then) http://math.ucr.edu/home/baez/week228.html there is a math trick you can do with the ISBN number of any book (involves numbers MOD 11, so there is a bit of algebra) (or if you think of adding numbers mod 11 as group theory then it has a bit of that) and more interesting, there is a game that two people can play with two pieces of rope It involves "rational" tangles or knots, and a number system that describes the knots the ga…

If the elements v1 v2 ... vn form a basis for the vector space V, then the elements must span V and be linearly independant. Also, the number of elements must be equal to the dimension of the vector space. So my question is this: If the number of elements is equal to the dimension of the vector space, and they are all linearly independant, will they always span V? I feel pretty certain that they will, but I am not completely sure. If they don't, what are some examples where they would not span but still be linearly independant?

these just posted on arxiv in the Representation Theory and Category Theory departments. Anyone we know? http://arxiv.org/abs/math.RT/0601578 Bousfield localization in quotients of module categories Matthew Grime 11 pages Representation Theory; Category Theory "We examine various triangulated quotients of the module category of a finite group. We demonstrate that these are not compactly generated by the simple modules and present a modification of Rickard's Idempotent Module construction that accounts for this. When the localizing subcategories are sufficiently nice we give an explicit description of the objects in the Bousfield triangles for modules that are …

Why is algebra so difficult? Other students in my class seem to have far less difficulty with it then I do and other math subjects are so much easier in comparison. I have trouble remembering all the theorems relating properties between definitions. I can't visualize anything, and I do practice alot. If I don't look at it for one week I already forget all kinds of stuff, while I can remember most things from other topics easily. I can follow the proofs in the book, but mostly by checking the logic, I can't see (and thus don't understand) the significance or the consequences of what has been proved. Any tips on how to make my algebra-life easier is greatly appreciated. …

Hi everyone! This seemingly easy question is proving difficult. I have to show that a field automorphism of R to R (real numbers) is the identity. What I've done: Let f:R -> R be an field automorphism. Since f(1)=1 is a generator of Z, f is the identity on Z. Because multiplicative inverses are sent to their inverses: f(1/x)=1/f(x), it follows that f is also the identity on Q. I`m not even sure if I`m on the right track. I obviously have to use some property of the reals now. A previous exercise which I very likely need to use asks to show that if x>0 then f(x)>0. Any help is appreciated.

First of all, I'd like to apologize for how softball this question may be to this particular part of the forum, but I'm in a linear algebra course and I'm having difficulty with this particular part. There is a matrix where A (A^-1) = I. I understand that, but I'm having difficulty finding a method for finding (A^-1). I thought I found one, but I worry that my book has a typo that throws the entire thing off. Any help on this would be greatly appreciated, and I'd like to mention I'm not just trying to leech, I've looked and can find nothing on methods, just the beginning and the end. Thank you.

At first reading this, it may not appear that this post really belongs in a mathematics forum, but on closer attention it may be found that its a very appropriate place for this seeming paradoxial word problem. Its nothing new, just (as far as I know) still mathematically/logically unanswered. I would prefer it to be spoken for or against on strictly mathematical terms, but verbal logic may also help. First assumption: Paradoxes cannot exist. Human misunderstanding on logic/truth can. Second assumption: Time travel is possible. Third assumption: There will be at least more than one moderator considering deleting or moving this post. No doubt most are …

My friend asked me to help him on his math for his economics, but I got rather lost since I do not know too much about Linear Algebra. His problem was that he had an economy (this is, for those who are curious, from Piero Sraffa) such that the inputs equal the outputs. The economy was: 280 qr. Corn + 12.t. iron --> 400 qr. corn 120 qr Corn + 8 t. iron --> 20 t. iron The solution to me was simple, just subtract from both sides the corresponding units to derive the exchange values of 10 qr. Corn = 1 t. iron. That much was correct, but I think by coincidence. Where my friend and I got lost was with a surplus. Suppose the economy became 280 qr. Corn + 12.t. …

My head is spinning from Penrose's treatment of twistors. Isn't it essentially a special sort of spinor? [math]Z^{/alpha} = (\omega^{A}, \pi_{A'})[/math] where Z is a twistor. Also, why is the linear and angular momentum used? Can't I use, say, something else that satisfies: [math] \omega^{A} = i r^{AA'}\pi_{A'}[/math]; [math]\frac{\omega^{A}}{\pi_{A'}} = ir^{AA'}[/math] or am I on the wrong track totally?

I want to solve a problem, to help me recall some more linear algebra. I want to solve a matrix system, and I want the answer to be a straight line in three dimensional space. So intuitively, i know this: If two planes are parallel they have absolutely no points in common, but If they are not parallel then they do have points in common, and the set of all points they have have in common lie on one and only one infinite straight line. Now, i know that the form of an equation for a plane in three dimensional space is: Ax+By+Cz=0 So choose two planes from the set of planes, but make sure they are not parallel. At this point, i want to solv…

anyone know the proof?