Linear Algebra and Group Theory

I don't know the procedure to factorize: [latex]a^2-ab+b^2-bc+c^2-ca[/latex] into [latex](a+\omega b+\omega^2 c)(a+\omega^2 b+\omega c)[/latex] [latex]\omega[/latex] is complex cube root of unity: [latex]\omega^3=1[/latex] ============================================= Can all the quadratic forms be factorized with complex roots? [latex]ax^2+2fxy+by^2+2gyz+2px+2qy+d=0[/latex]

This is a rather unusual "theoretical" question. I was thinking of basic arithmetic..of a certain kind and was wondering if there is (probably) an area of theo maths or physics that covers it. The way we view the world as far as counting is concerned is that there are 2 ways to go, up or down. For example from number 3 i get to 4 by adding 1 or i get to 2 by subtracting 1. This is the definition of addition in the simplest form. (subtraction can be argued is addition of negative numbers) Is there a 'space' where there are 3 ways to go, so that from element 'a' we can go to 'b', 'c' or 'd'? Adding 1 wouldn't really make sense then, or it would have to be defin…

I never understood this concept. What proof backs this up? This argument is sometimes also presented as "2 + 2 = 5 for large values of two." This doesn't make sense to me whatsoever. Can someone please help?

Hi there, I haven't been on here for almost a decade but I was wondering if someone could have some ideas on a binary (ternary would can also be acceptable) , discrete metric which is intransitive, that is fully, not partly, i.e. for all x,y,z: xRy ^ yRz => !(xRz). I see various ideas in topologies which are not dense and so on but it must be a discrete metric function. I been brainstorming for a good while but I can't quite come up with a solid solution, so ideas are very welcome.

My math teachers say that 0 to the power of 0 is undefined, but after messing around a little, I found this: 00=00 Original equation 00=00*1 Multiply one side by one 00/00=1 Divide both sides by 00 00=1 Simplify using the division laws of exponents Are my teachers incorrect or did I make a mistake?

I am a bit confused by this. Here is the equation: [math] a=b[/math] [math]a^{2}=b^{2}[/math] [math]a^{2}-b^{2}=0[/math] [math](a+b)(a-b)=0[/math] [math]a=b,-b[/math] let [math]a=1[/math] [math]1=1,-1 [/math] I understand that this cannot be true but why does it work algebraically? To my understanding, if [math] a=b [/math], than [math] a^{2}=b^{2}[/math] but if [math] a^{2}=b^{2} [/math], than [math] a\neq b [/math] What am I not understanding?

i keep seeing this problem and i dont see how it posible

We had our first test and I'm trying to understand what it is I'm missing. I list each of the questions below, followed by what I answered. 1. Let v1=[ 1 ] and v2 = [ 1 ] --- Find a nonzero vector w that exists in R^3 such that {v1, v2, w} is linearly independent. [ 1 ] [ 2 ] [ 1 ] [ 3 ] ans: w = [ 1 ] this was assuming that as long as the vector was a multiple of the of the vectors then the set would be linearly independent. [ 4 ] [ 6 ] 2. Find the general solution to the equation A*x = 0 (where x is a vector). Give your answer in parametric vector form. A = [ 1 2 0 -2 0 ] [ …

I think that 28 is such a number. Do you agree? I also think that it is the smallest such number. Do you agree with this?

Hi, there. I have a calculation as following: BCA+B'CA'+B"CA"+... where A,B,and C are square matrices. A and B are symmetrical matrices. Since C remains the same in all the terms of this calculation, I am wondering is there a easier way to carry out this calculation, e.g. make the multiplication only once like the form of: (B+B'+B"+...)C(A+A'+A"+...) I know this form doesn't work, and it is silly to ask such a question, but just in case any one can light a light for me to some other easier forms. Thank you very much!!!

Hi, I know how to compute determinants and I'm familiar with the geometrical meaning of determinant as the scaling factor of a unit (point/square/cube/hypercube)'s area/volume by applying a linear transformation (using a matrix). However, I have several questions: Let's say I define determinant to have the above meaning. How can one derive the formula for computing determinant following just the visual/geometrical meaning? Let's say I have an arbitrary closed 2D polytope [latex]P[/latex] and I transform all of its vertices by a matrix [latex]\mathbf{A}[/latex]. Is [latex]\det\left(\mathbf{A}\right)[/latex] the scaling factor of polytope's [latex]P[/latex] area afte…

Since $\mathbb{R}$ is not first order definable, the statement $\forall x \in \mathbb{R}(x = x)$ is not a first order statement and thus not provable in ZFC. Does that mean we can assume $\exists x \in \mathbb{R}(x \neq x)$? If so, would this provide us with the basis for a field with one element?

Hello, In algebra and number theory, it can be demonstrated that 0^0 = 1 is true, but in analysis, 0^0 is an indeterminate form. Through the calculation of limits, it can be equal to 1, to other finite values, diverge, or even not exist. This is why, when faced with this issue, mathematicians have conventionally set 0^0 = 1. So, let's be bold: instead of saying that 0^0 does not have the same value in all contexts and that 0^0=1 is a convention, let's change the notion of the number 1 so that 0^0 = 1 in all contexts. In this new conception of the number 1, it would be both a number and an indeterminate form, meaning that it could be equal to 1, to ot…

We have: P(A|B) = P(B|A)P(A)/P(B). Is P(B) = P(B|nothing) or is P(B)=P(B|everything)?

Is this the correct folder in which to pose a question on differential topology? To help keep things simple, suppose you have a 2-form on a manifold of dimension 2, having metric type (+, +), rather than an arbitrary number of dimensions and other metrics. As I understand it, we can integrate a k-form over an n dimensional manifold where k=n. But I want to do an incomplete integration: Given, L an antisemmetric tensor, [math]L=L_{[ij]} dx dy [/math] define [math]S = \int L = \int L_{ij}dx dy[/math]. However, I would like to partially integrate L. [math]S_y dy = ( \int L_{ij}dx ) dy[/math] and …

Let u and v be any vectors in a vector space V. Prove that for any vectors xsub1 and xsub2 in V, span{xsub1, xsub2} is a subset of span { u, v} if and only if xsub1 and xsub2 are linear combinations of u and v.

I am currently studying Turing machines and I have set myself the following problem Using Pythagorian theory 1 Adjacent side = 12 inches Opposite = 12 inches Therefore hypononuse =19.7 (approx) 2 Adjacent = 1 foot Opposite = 1 foot therefore hypotonuse = square root of 2 which is a non computable 'irrational number' Same triangle - same dimensions First example - computable, second non computable - but the are the same. Go easy on me - my maths is not great... simple explanation appreciated. Zero

I am not a mathematician, but rather a musician who met a problem (it's not homework!) that can possibly be posed as a mathematical problem. So I'm looking for a solution but also and especially for a direction to the discipline that may solve the problem. I apologize for bad use of proper mathematical writing. Let's say I have the whole set of all 64 possible 6-tuplets of digital bits, such as 000000, 000001, (000010) and so on up to 111111. Of course I can operate on couples of 6-uplets in order to have the XOR result. For example: 010101 XOR 11000 = 011010. I would like to put the 64 6-uplets in such an order (a cyclic one with period 64, that is to say in circle…

Hi Everyone, I just learnt this game and ever since have been thinking about it. But couldn't figure it out with that garbage bag over my shoulder. So, I decided to post it here, but was a bit confused with where to post it. So, here it is - 1. First take 52 cards. 2. Tell Someone to think of a card in his mind. (Don't seperate the card, just think of it) 3. Then make three sets of card - I mean two sets will have 17 cards and one will have 18. (Make the sets like when u distrubute cards when u r playing a game , don't just take 17 , then 17 and then 18. Distribute them) 1 1 1 2 …

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…

Hi, Is there any solution for the following problem: [math]Ax = \lambda x + b[/math] Here [math]x[/math] seems to be an eigenvector of [math]A[/math] but with an extra translation vector [math]b[/math]. I cannot say whether [math]b[/math] is parallel to [math]x \quad[/math], ([math]b = cx[/math]). Thank you in advance for your help... Birkan

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

Where to see proof of the fact that the linear space is left free module over the ring?

Hi, I have three n*n matrices: A, B, and C. They are all real. A and B are symmetrical but C is not. M=AC+CA; and E=BM+MB. Based on the answers of the question before (the one about the M=AC+CA), I think there is no way to make M by one matrix multiplication. Thus I think the generation of E should be two matrix multiplications as well. This matrix multiplication will be carried out once for every loop of the program. Since I found the matrix C is a constant matrix for all the loops, I wonder is there a way that I can make the matrix multiplications once for all the loop, such as E=XC+CX or something like that? Thank you so much.

hello, so i'm currently working on a linear programme optimisation, and i was wondering if someone could help me with a bit of maths. and i need to represent abs(x) in terms of a linear equation (so no root(x^2) allowed) everyone knows what absolute(x) or |x| represents and it's pretty simple to do in your head, but I was wondering if there were any computational methods or equations for evaluating this? also i'm looking specifically for a linear method so i cant have if statements and such also are there any functions that result in x = x if x>=0, x = 0 if x<0 if so what is it called? any help is appreciated, thanks