Linear Algebra and Group Theory

Hi all! I'm working on the following problem: If the ciphertext message produced by RSA encryption with the key (e, n) = (5, 2881) is 0504 1874 0347 0515 2088 2356 0736 0468, what is the plaintext message? My work thus far: 2881 = 43 * 67 phi(2881) = phi(43 * 67) = phi(43) * phi(67) = 42 * 66 = 2772 From here I need to find an inverse of 5 modulo 2772 which is 1109 and then raise each ciphertext block to power 1109 mod 2881 to retrieve the plaintext message. I don't know how to do this last step... 0504^1109 = ____ (mod 2881) How to fill in this blank above? Please advise. Thanks!

In case anyone here hasn't noticed, there have been a lot of debates on this online between people on other forums, some with a good grasp of mathematics and some without. The debate is over whether 0.999 does or does not equal 1. There are many different proofs to prove that it does, but because some people understand some proofs better than others, and some not at all the debates still go on. I've personally been involved in these sort of debates and I can think up at least five different proofs, all based on different sorts of logic, that show that they are equal. The more forms we can find, the better the chances are that someone will understand it. I thought …

let's say that their is a linear transformation T:V --> W. If X1,...,Xr are vectors in V such that T(X1),...,T(Xr) are linearly independent, then is it necessarily true that the vectors X1,...,Xr are linearly independent as well?

hello all, Can anyone here tell me how to get the same row echelon form by hand and using the ti-86? For example, I am trying to find the basis for the row space of matrix A and when I do the operations by hand I get one set and when I do the operations on my ti-86 I get a different set. According to the solutions manual that accompanies the text, the result calculated by the ti-86 are wrong. Any suggestions? thanks in advance.

(i). Let G be a simple connected cubic plane graph, and let phi_k be the number of k-sided faces. By counting the number of vertices and edges of G, prove that: 3*phi_3 + 2*phi_4 + phi_5 - phi_7 - 2*phi_8 - . . . = 12 (ii). Deduce that G has at least one face bounded by at most five edges.

First question of the first exercise sheet from my graph theory course, and I'm just stuck. Maybe I am not thinking right. Any hints are welcome Questions: Show that in any group of 6 people, either there are 3, each one of whom knows the other two, or there are 3 people none of whom knows the other two.

Say I have a set X and a topology T on X = {X {} A} i.e A is an open subset of T. Then the complement of A is X - A = Ac, which is closed. Now the interior of A, int(A) is the largest open set (or the union of all open sets) contained in A which is A, and the closure of A, cl(A) is the smallest closed set in {X {} Ac} containing A which is X. So if bd(A) = cl(A) - int(A), we have that bd(A) = X - A = Ac. Similarly, the closure of Ac is the smallest closed set containing Ac, which is Ac = X - A. So, by an alternative definition of the boundary of A, cl(A) intersect cl(Ac) = X intersect (X - A) which is X - A = Ac. I've tried it out on a number of arbitrary…

Is the set of functions: {[math]f® = R |\frac{df}{dx} + 2f = 1[/math]} a vector space? I said no because it doesn't seem to have a zero vector, but I'm doubtful of my answer. Can someone help me prove its vector space validity (or lack thereof)?

I am really stumped on this one, I have been working on it for nearly 2 weeks now and I am beginning to wonder if it is even possible. So I though I might throw it out there because there are a whole lot of people that are alot smarter than me that might be able to figure it out. So lets say you have A,B,C, and D all integers 0 - 1000. Now I am trying to find an equation that will allow me to take A,B,C and D and come out with A,Z, and D. But the hard part is I have to be able to take A, Z, and D and come out with the original B and C. Any help at all would be greatly appreciated. Thanks

Can some please explain to me why [math]\{a + b + c = 0 | a,b,c \in R^3\}[/math] is a vector space but [math]\{a + b + c = 1 | a,b,c \in R^3\}[/math] isn't? And how do I get the {} to show up?

If G is an abelian group and n>1 an integer, let A={a^n such that a E G}. Prove that A is a subgroup of G. isn't the identity of A a^0 which does not fall under n>1

Given two numbers a and b (a, b E R), use indirect proof to prove that a^2 +b^2 is greater and equal to 2ab. Any ideas? Any help would be greatly appreciated. Kev

I am new here, so I do not really understand how this forum thing works yet, so bare with me. When you are identifying whether something is a subgroup of a group (I can do that part), how do you give a "complete list of all subgroup relations?" I keep thinking that this is the skeleton diagram. i.e. Z4 - {0} - {0,2} That is an example that our professor gave in class, but I am not sure that I understand that very well either. I do, however, understand that Z4 is modulo 4, but how does the {0} and (0,2} get pulled out? There are many more relations.

I am sorry to say that I do not understand the metric tensor one bit. Isn't it the Kronecker Delta in Euclidean space? And wouldn't that be a collection of row vectors?

I have read probably over 100 different books on set theory over the past 15 years. I have a question, it's about notation and that's all it is about. Suppose that someone uses the following notation for a set: {a,b,c} Must I infer that the set has three elements in it OR have they left open the possibility that the set contains one element, or possibly even two elements? I could adjust my logical structure to accomodate anything, but the question is only about standard usage of the notation above. Thank you

Hey all. Something great occured to me this morning. I've done a little digging and as far as I know this has not been done yet. I would like to build a strange attractor for prime numbers. I thought one axis would be the distance between prime n and prime n+1. I'm having trouble deciding on the the other two axis and thought you all might have some input.

I'm thinking of buying a number theory book for self-study. I was wondering.. what are the prerequisites in math subjects for successfully learning number theory? (besides arithmetic ) Also, any good book recommendations would help. Thanks

If X is a linear continuum space how do we show that all convex subsets of X must inheret the linear contiuum axioms, that is, they have betweenness and least upper bound property. Thanks.

hello: can somebody tell me the real world application of trinogometry,number theory thanks for any tips.

How can I show the set of integers Z cannot be made into a vector space over R? Z=(0, +-1, +-2,...) For example I had to show how the set of integers Z cannot be made into a vector space over C by: (i) exits in V and a vector (1) exits in Z i*1=i therefore, i does not exists in Z Hence it is not a vector space because it does not close under scalar mulitplication. I have to do the same with rationals (Q) and with reals®

Can a vector be a wave? Or does the wave need to be a number of vectors?

Could anyone give me a brief exposition, or direct me to books/resources, on spinor algebra/calculus? Thanks in advance

so, I've been reading up on alot of linear algebra lately, and I've been seeing alot of uses for determinants. I learned how to calculate using them and all that good stuff but, I have not been able to figure out what it is exactly, is there some sort of geometric representation of it or anything else like that?

What is the sum of the first n Fibonacci numbers? Use the Fiboncci numbers below to make a conjecture, and then prove it using mathematical induction. [b]n [i]tn[/i][/b] 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 ..... I know that each Fibonacci number is the sum of the previous two, but I am unsure of how to do the question above. Could someone please show me?

What are the axioms of a group? I suspect I know them, but i've read conflicting answers from time to time.