quotient Definition and 356 Threads

So I have an elementary understanding of group theory and the goals of studying sets and operations on them but I noticed that along the way, my understanding of a quotient group is severely flawed. I understand what a so called normal subgroup is, but could someone please give an in depth...

Problem: Let L and M be finite dimensional linear spaces over the field K and let g: L\times M \rightarrow K be a bilinear mapping. Let L_0 be the left kernel of g and let M_0 be the right kernel of g. a) Prove that dim L/L_0 = dim M/M_0. b) Prove that g induces the bilinear mapping g': L/L_0...

Homework Statement Show that if a (is in) Z and d (is in) Z+, d>1 then the quotient and remainder when a is divided by d are a/d and a-d(floor function(a/d)) Homework Equations The Attempt at a Solution solution (that i have from handout - that i don't understand) by thm 2 p202 (? i am not...

So earlier this year I came here to discuss about having fun with groups, rings and isomorphisms and such. I fell upon the idea of finding an isomorphism of the positive rationals to the sequence of the exponents found in their prime factorization. I didn't know what much to do with it since I...

EDIT: I found the mistake, question is answered! Its funny because I spent 40+ minutes trying to get the right answer and looking for the mistake but typing it all out in latex helped me to find it! Homework Statement \frac{\sqrt{x}}{x^3+1} The Attempt at a Solution...

Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...

Hey all, We have not covered quotient vector spaces in class, but my homework (due before next lecture) covers a few proofs regarding quotient spaces. I've done some reading on them and some of their aspects, but as it is still a new concept, I am struggling with how to go about this proof...

Homework Statement http://images.calcchat.com/solutionart/etf5e/03/c/se03c01047.png Homework Equations The Attempt at a Solution What am i missing, because i get: 6/4 [(-1 + tanx*secx- (tanx)^2]

Let X be the quotient space obtained of \mathbb{R} identifiying every rational number to a point. Is X a Hausdorff space? Is X a compact space?

Homework Statement Let \mathbb{CP}^n be n-dimensional complex projective space, and let \pi: \mathbb C^{n+1}\setminus\{0\} \to \mathbb{CP}^n be the quotient map taking \pi(z_1,\ldots,z_{n+1}) = [z_1,\ldots,z_{n+1}] where the square brackets represent the equivalence class of lines through...

If X is a topological vector space and Y is a subspace, we can define the quotient space X/Y as the set of all cosets x + Y of elements of X. There is an associated mapping \pi, called the quotient map, defined by \pi(x) = x + Y. If I'm not mistaken, there is an equivalence relation lurking...

Homework Statement I am having trouble with a couple difference quotient questions. Here is a question I can't seem to solve. Evaluate the difference quotient for the given funtion and simplify your answer. f(x) = 4 + 3x - x^2 , f(3 + h) - f(3) / h Homework Equations...

Homework Statement This is from Lee's Introduction to Smooth Manifolds. Suppose π : X → Y is a quotient map. Prove that the restriction of π to any saturated open or closed subset of X is a quotient map. Homework Equations Lee defines a subset U of X to be saturated if U = π-1(π(U)). π...

Find d^2y/dx^2 when dy/dx = (3x^2 - 24x - 45) ÷ 2y i tried by (6x-24) ÷ 2y. Unsure what to do about the y on the denominator.

Homework Statement Find dy/dx for the following function: y = (11-cos(x))/(2+cos(x)) Homework Equations Quotient Rule: y'= ((g(x))(f'(x)) - (f(x))(g'(x)))/ (g(x)^2) The Attempt at a Solution I used the quotient rule to come up with this: y'= ((2+cos(x))(sin(x)) -...

I'm curious why people develop these objects. Although I've seen some proofs of theorems using coset(or quotient space somtimes), it remains mysterious to me how people come up with these in the first palce. So what's the motivation for inventing coset or quotient space, logical and historical?

I have a question I need to resolve before my exam on thursday. It relates to the following result: Let N be a normal subgroup of G, and let K be any subgroup of G containing N. Then K/N is a subgroup of G/N. Furthermore, K/N is normal in G/N if and only if K is normal in G. The first part...

Homework Statement S = (8x^2 - 7x + 125)/(8x+7) Where the value of x is unknown The Attempt at a Solution u = 8x^2 - 7x + 125 du/dx = 16x - 7 v = 8x + 7 dv/dx = 8 ds/dx = (v*du/dx - u*dv/dx)/v^2 = (8x+7*16x-7 -...

Homework Statement Remember to show your working explicitly throughout your answer to this question. (a) (i) Use the Composite Rule to differentiate the function f(x) = (x^2− 6x + 23)^(3/2) (ii) Use the Quotient Rule and your answer to part (a)(i) to show that the function: g(x) = (x −...

Homework Statement Let b > a > 0. Consider the map F : [0, 1] X [0, 1] -> R3 defined by F(s, t) = ((b+a cos(2PIt)) cos(2PIs), (b+a cos(2PIt)) sin(2PIs), a sin(2PIt)). This is the parametrization of a Torus. Show F is a quotient map onto it's image. Homework Equations Proving that any subset...

When differentiating the following equation, y=\frac{d}{dx}\frac{e^x}{x}, why is it wrong to rearrange the equation to y=\frac{d}{dx}\frac{1}{x}e^x and apply the product rule? Doing so gives me a different result than using the chain rule in conjunction with the quotient rule. Sorry about...

Does quotient space V/N include N itself? I think it does because: V/N includes all the cosets of N, and a coset of N is defined as N+\alpha={\epsilon+\alpha: \epsilon \in N}. This definition is from Advanced Calculus by Loomis & Sternberg and it does not say \alpha cannot be 0. So V/N...

Homework Statement Part of a larger problem. I know that F_{1}^2+2F_{1}F_{2}-F_{2}^2=0 where F_{1} and F_{2} are x and y components of a force. Hence \frac{F_{1}}{F_{2}}=1\pm\sqrt{2} I can't see how that step is done. Homework Equations The Attempt at a Solution...

Hi, given a polynomial ring R=\mathbb{C}[x_1,\ldots,x_n] and an ideal I=\langle f_1, f_2 \rangle, \quad f_1, f_2 \in R, is it always true that R/I \cong (R/\langle f_1 \rangle)/\phi(\langle f_2 \rangle), with \phi: R \rightarrow R/I being the quotient map? That is, is quotienting by I always...

Hi all, I'm a beginner in calculus so my question might be stupid. When a function is differentiable, then in difference quotient one can always factor 'h' out in the numerator, even if the function is exponential and 'h' is in the exponent. Is some magic behind this or something else? I've read...

Homework Statement The Derivative of x-2/(x^2-1) Can you show me how to get the derivative using the quotient rule and tell me if the order matters? I always thought that the order never mattered until I got a wrong graph because it seems I got the wrong derivative. Homework...

Hi All, I've come across a theorem that I'm trying to prove, which states that: The quotient group G/H is a discrete group iff the normal subgroup H is open. In fact I'm only really interested in the direction H open implies G/H discrete.. To a lesser extent I'm also interested in the H...

Here is theorem 9.2 from Stephen Willard's General Topology: If X and Y are topological spaces and f:X\to Y is continuous and either open or closed, then the topology \tau on Y is the quotient topology induced by f. So f has to be onto doesn't it? Otherwise there will be multiple...

In my calculus book, the method of increments is used to find the derived function of the product and quotient of two functions. For example for the derivative of the product of functions u and v where u0 and v0 are the values of u and v at x = x0: y = uv y' = u0(dv/dx) + v0(du/dx) +...

dy/dx - a limit or a quotient?? Hello Friends. I have a confusion in differentiation. We know that lim \Deltax->0 ; \Deltay/\Deltax = dy/dx. Then how can dy and dx be treated as independent variables, and dy/dx as their quotient, if dy/dy is actually a limit? Also, since we won't be able to...

Homework Statement (x)/(x^2+1)^1/2 Homework Equations The Attempt at a Solution I got to this (x^2+1)^-1/2[(x^2+1)-x^2/(x^2+1)

How do I find the indefinite integral \int\frac{dx}{x^{2}+16} without using partial integration or variable change? I have no clue how this can be done.

Let X and Y be topological spaces; let p:X -> Y be a surjective map. The map p is said to be a quotient map provided a subset U of Y is open in Y if and only if p^-1(U) is open in X. Let X be the subspace [0,1] U [2,3] of R, and let Y be the subspace [0,2] of R. The map p:X -> Y defined by...

Homework Statement Show that if f(x)=sinx then (f(x+h)-f(x))/h=((sin(h/2))/(h/2))(cos(x+h/2) Homework Equations Trig identities, possibly the half angle formulas? The Attempt at a Solution (f(x+h)-f(x))/h = (f(x+ h/2 + h/2)-f(x))/(h/2 + h/2) = (sin(x+ h/2 + h/2)-sin(x))/(h/2 +...

Homework Statement Show that if f(x)=sinx then (f(x+h)-f(x))/h=((sin(h/2))/(h/2))(cos(x+h/2) Homework Equations Trig identities, possibly the half angle formulas? The Attempt at a Solution (f(x+h)-f(x))/h = (f(x+ h/2 + h/2)-f(x))/(h/2 + h/2) = (sin(x+ h/2 + h/2)-sin(x))/(h/2 +...

Homework Statement derive f(x)=x2+2x+1/ x2-3x+2The Attempt at a Solution (x2-3x+2)(d/dx)(x2+2x+1)-(x2+2x+1)(d/dx)(x2-3x+2)/(x2-3x+2)2 =(x2-3x+2)(2x+2)-(x2+2x+1)(2x-3) =(2x3-4x2-2x+4)-(2x3+x2-4x-3) =5x2+2x+7/(x2-3x+2)2 that's what I'm getting... but according to the online derivative...

In Lectures on the hyperreals: an introduction to nonstandard analysis, pp. 50-51, Goldblatt includes among his hyperreal axioms that the sum of two infinitesimals is infinitesimal, that the product of an infinitesimal and an appreciable (i.e. nonzero real) number is infinitesimal, and that the...

Hi All. I have several questions on abstract algebra. Here are my questions and the attempts I had done so far. (1)Let denote Z as the integer.According to theorem, the direct product of Z3 X Z7 = Z21. Hence, is Z4 X Z2 is equal to Z8? Z2X Z2 is equal to Z4? (2)For Z4 ={0,1,2,3 }, and Z2...

I have been putting some thought into understanding Gibbs energy but I can't quite figure one thing out. Here is my dilemma: Say that someone wants to react A with B to form C, and they mix pure A with pure B. At the moment the reaction starts, there is no C in the mixture (is this...

Homework Statement find f(a+h)-f(a)/h for the f (x)= 2x+3x2 Homework Equations f(a+h)-f(a)/h The Attempt at a Solution f(a+h)-f(a)/h =2(a+h)-3(a+h)2-2x+3x2 =2a+2h-3a2-3h2-2a+3a2 =2h-3h2/h that is where i am stuck help please

Homework Statement Given the function , calculate the following values f(x) = 2-5x f(a) f(a+h) f(a+h)-f(a)/h The Attempt at a Solution f(a)= 2-5x f(a+h)= 2-5(x-h) f(a+h)-f(a)/h = -5 i am 100% sure but the darn computer is telling me that f(a) and f(a+h) is wrong. my calc professor is using...

I am taking an online Introductory Calculus course. I have a decent understanding thus far, however, the problem I'm working on gets somewhat messy and I am having a difficult time simplyifing the answer. f"(x) = (x^2 + 9)^2 (-2x) - [(9 - x^2)(2)(x^2 + 9)(2x)]/(x^2 + 9)^4 the solutions manual...

Homework Statement Find the derivative of f using the difference quotient and use the derivative of f to determine any points on the graph of f where the tangent line is horizontal. f(x)=3x^3-9x Homework Equations The Attempt at a Solution \lim_{\Delta...

we know there is a two to one homomorphism from SU(2) to SO(3) suppose u is an element in SU(2) then u and -u map into the same element in SO(3) the question is, maybe SO(3) is a quotient group of SU(2)? with respect to the subgroup {I,-I}?

If you can write, for example \frac{cosx}{x+1} = (cosx)(x+1)^{-1} then what is the point of the quotient rule? Can't you just use the product rule for everything?

Specifically, how do you prove the quotient rule using a similar method that Leibniz used for the product rule?: http://en.wikipedia.org/wiki/Product_rule#Discovery_by_Leibniz I've tried it once for d(u/v) but I keep getting a vdv term in the denominator.

Homework Statement Let G be a finite group and N\triangleleftG such that |N| = n, and gcd(n,[G:N]) = 1. Proof that if x^{n} = e then x\inN. Homework Equations none. The Attempt at a Solution I defined |G| = m and and tried to find an integer which divides both n and m/n. I went for some X...

As a way to keep busy in between semesters I decided to work my way through Algebra by Dummit and Foote in order to prepare for the fall. Working my way through quotient groups is proving to be quite difficult and as a result I'm stuck on an exercise that looks simple, but I just don't know...

Hi, everybody: I have read different sources for quotient bundle: Milnor and Stasheff, Wikipedia Wolfram, and I still cannot figure out how it's defined. All I know is that it involves a space X, a subspace Y, and a restriction. Let Tx be a bundle p:M-->X for X, and...

\int \frac{1}{2x+3}=\frac{\ln |2x+3|}{2}+c so why is \int \frac{1}{x^2+x}\neq \frac{\ln |x^2+x|}{2x+1}+c ? is it because in general , \int \frac{1}{x}=\ln |x|+c the denominator is meant to be only linear function ?