Linear Algebra and Group Theory

I want this thread to be about Rings. I think the main issue for the thread should be, what is/are the purpose/purposes of defining a ring. Let me start off with the definition Dave gave, of a ring, in a different thread. A ring is a set R, with two operations defined on it, which are symbolized +, *. The ring R, must be a group under at least one of the two operations. It doesn't have to be a group under both, but it could be. Suppose that the ring is a group under +. That will mean the following four things are true about elements of R: 1. There is a zero element of R, denoted by 0, such that for any element x of the ring, 0+x=x. 2. Associativ…

Any suggestions? Online? Textbooks? Rev Prez

What is SO(3)?

I get stuck on the question " for any simple ring (no 2-sided ideal except 0 and itself) has a unique faithful irreducible module upto isomorphism". can u pls help me to work it out? many thanks

OK, firstly I hope this is the rigth place for my question. I'm in a bit of a problem. I need to be able to calucalte the Killing for for a Lie algebra by next wek, but I'm stuck and won't be able to get any help in 'real life' until Friday, not leaving me enough time to sort out my problem. So I was hoping someone here might be able to show me some pointers. So I have the Lie algebra of all upper triangular 2x2 matrices and am using the basis (1 0) (0 1) (0 0) (0 0),(0 0),(0 1). (I hope these matrices turn out OK: they should be 3 2x2 matrices.) The Killing form is defined as K(X,Y) = trace(adX adY) for all X,Y in the Lie algebra, where ad is the adjoint, d…

After talking with Rev, it became apparent, that I need to review linear transformations. Last night, I spent about 4 hours reading through 3 different books on linear algebra. All I want to do right now, is fully and competently understand what a linear transformation is. Here is where I am at, and would appreciate any help. Ok first of all... The first thing that became apparent to me, was that I need to be able to logically express the concept of "exactly one," as opposed to merely "there is at least one." And I am a stickler about my logic. So for right now, I would like know exactly how some of you address the issue of "exactly one" using symbol…

how do you read [math]R(f,P)[/math] I know that P is the partition and that it means the same thing as [math]\sum_{j=1}^k{f(x_j)}[/math], but what is f? in [math]\sum_{j=1}^k{f(x_j)}[/math], i think j is the lower bound and k is the upper. is that the case? like j is the start and k is the end. well, that is how i use it and it works (at least for the Mercator series)

Part of my course im doing at the moment is looking at how to determine the inverse of a matrix but i just cant seem to get it, could someone explain how this is done, not in the ways by reducing the to an identity (as i know how to do this) but the proper way where a colomn and a row are deleted.......and something could someone explain this in lamens to me pls? Heres a Matrix to use as an example C=(1 5 3,4 4 3,2 1 1)

If we have X with order topology, and we have [a,b] subset of X. And if we know that we have U open in X, can we conclude that [a,b] /\ U is open in [a,b]? If so please explain.

http://math.ucr.edu/home/baez/week212.html a new issue of "TWF" came out today it seems there are only 4 possible "normed division algebras" in Baez terminology and they are the reals, the complexnumbers, the quaternions, and the octonions and it turns out that by introducing a kind of Z2 symmetry or a very simple kind of twopart boolean symmetry you can EXTEND this and get more nice number systems which are analogs of the "normed division algebras" I am not sure i have the terminology right, but it belongs in mdrn algebra and also in physics because these new systems are called superalgebras, or else "super division algebras" and they are like the …

Hi The problem im having is I have a school which is made up of Instructors & Pupils, A Pupil can only be in one lesson at any given time. Im not sure how do I express this in Z? Do I inteduce variables say Max_Lesons_At_Given_Date_Time, and Lessons_At_GivenDate_Time_Per_Pupil. And say something like: Max_Lesons_At_Given_Date_Time = 1 #Lessons_At_GivenDate_Time_Per_Pupil ≤ Max_Lesons_At_Given_Date_Time Or do I just use one and say: # Lessons_At_GivenDate_Time_Per_Pupil ≤ 1 Any ideas thanks

Does the summation of reciprocal prime pair products 1/(2*3)+1/(3*5)+1/(5*7)+1/(7*11)+1/(11*13)+1/(13*17)...=? or the summation of reciprocal Fibonacci pair products 1/(1*1)+1/(1*2)+1/(2*3)+1/(3*5)+1/(5*8)+1/(8*13)+1/(13*21)...=? have any significance in number theory?

ok nevermind my request

Hello all..I am really having HARD TIME working the following problem..it took me forever and I am about to give up..I need help or even a hint on how to solve it..it is linear algebra/finding eigenvalues' problem. Let V = R[X]_2 be the real polynomials of degree at most 2. Let T ( a linear transformation from V to V) be defined as (Tf)(x) = f(x) + f '(x). Find all the eigenvalues of T and their geometric and algebraic multiplicities.

A coin is tossed until a head occurs and the number of tosses required is observed. What is the sample space? The answer is [math]\Omega = \{1,2,3,...\}[/math] and my lecturer said, "one, two, three, till infinite." But if you flip a coin inifite times, wouldn't you expect heads to come up half the time...?

I would really appreciate it if someone can check the proof to a claim i made in a bigger proof..here is the whole thing and below it is the part i want you to check: Let X be a space, Y subset of X, and Z subset of Y. Give Y the subspace topology the following hold: (1) Suppose Y is closed in X. Then Z closed in Y implies Z closed in X. (2)Suppose Y is open in X. Then Z open in Y implies Z open in X. Let us start with (2). Denote by T the topology on X. Then the topology on Y is {U \cap Y | U in T}, where \cap = intersection. Thus if Z is open in Y then Z = U \cap Y for some U which is open in X. But now U and Y are both open in X, so their intersection …

Suppose you have a base {e_1, e_2, e_3} in R_3. You don't know anything else about them. Is it possible, if you are given another base {u_1, .. , u_3} to give a change-of-base matrix from the e_i's to the u_i's??? Thanks!

Here is the question: If a group G has a composition series of length two, then for any two distinct normal subgroups M and N, G= M x N..How to see that? what I am thinking is whether we can get G>M>1 is a composition series.(every composition series should have length two from Jordan Holder) Thanks a lot! Simon

Suppose R is a PID which is not field, M is a finitely generated module on R. prove that if for any prime p, M/pM is cyclic R/pR module, then M is cyclic. I am just trying using the uniqueness of the structure theorem about finitely generated module over pid...but don't know how to connect M with M/pM. Any help would be appreciated. Simon

I know what I am trying to prove is very simple, but I am stuck and I need help.. In vector spaces, let V be a vector space over F, c in F and v in V. I am trying to prove the following: c0 = 0 My proof: I know the main idea here is to use 0 = v + (-v), but until this point, we don't know that -v = -1.v, we don't know that if v /= 0 then there is an inverse such that v*V^-1 = 1, so let's see..by wa of contradiction, assume c0 /= 0 then c(v +(-v)) /= 0 I am stuck here..any suggestions?

Any help on the following problem would be appreciated: Consider domain R= Q + x^2 Q[x] (polynomial without linear term) , prove that for every f, R/fR is a finite dimensional vector space on Q. and prove that every prime ideal in R is maximal~ Thanks Simon

So, my econ book explains that I should use a input-output matrix for a specific situation...but what the devil is a "input-output matrix"?

group of order 540 is simple~~ I get from sylow's theorem that r5 = 36, r3= 10, r2 = 45 ..I cannot get contracdiction with counting elements... Any help would be appreciated~ Simon

For some hint about the following problem, thx a lot~ Let R be a ring with 1. A nonzero left R -module S is simple if 0 and S are the only submodules of S . Let 0 ---> S---(alpha)--->M--(pi)-->S---->0 be a short exact sequence of R -modules which is not split, and such that S is a simple R -module. Show that the only nonzero submodules of M are alpha(S) and M Rp~

Question: Let F be a finite field of odd characteristic. Prove that the rings F[X]/(X^2 - a) as a ranges over all nonzero elements of F fall into exactly two isomorphism classes. I am thinking to classify into two cases: a is a perfect square in F and a is not a perfect square. But I don't know what the condition "the character is odd (shou be odd prime?!)" will do here. Thx a lot! Rp~