Product Definition and 1000 Threads

The inner product axioms are the following: ##\text{(a)} \ \langle x+z,y \rangle = \langle x,y \rangle + \langle z,y \rangle## ##\text{(b)} \ \langle cx,y \rangle = c\langle x,y \rangle## ##\text{(c)} \ \overline{\langle x,y \rangle} = \langle y,x \rangle## ##\text{(d)} \ \langle x,x \rangle > 0...

Homework Statement Let ##V## be a vector space equipped with an inner product ##\langle \cdot, \cdot \rangle##. If ##\langle x,y \rangle = \langle x, z\rangle## for all ##x \in V##, then ##y=z##. Homework EquationsThe Attempt at a Solution Here is my attempt. ##\langle x,y \rangle = \langle x...

Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product? Thanks! Andy

Homework Statement Let ##A## and ##B## be finite groups, and ##A \times B## be their direct product. Given that ##(a,1)## and ##(1,b)## commute, and that ##(a,1)^n = (a^n,1)## and ##(1,b)^n = (1,b^n)## for all a and b, show that the order of ##(a,b)## is the least common multiple of the orders...

Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...

Show that the product of the roots of the equation x^2 + px + q = 0 is q. I need help with the set up. Must I use the discriminant here?

In group theory, what is the direct product of a symmetry group with itself? Say T*T or O*O?

Homework Statement Consider G = {1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64} with the operation being multiplication mod 65. By the classification of finite abelian groups, this is isomorphic to a direct product of cyclic groups. Which direct product? Homework EquationsThe...

Homework Statement Show that ##g=g(x_1,x_2,...,x_n)=(-1)^{n}V_{n-1}(x)## where ##g(x_i)=\prod_{i<j} (x_i-x_j)##, ##x=x_n## and ##V_{n-1}## is the Vandermonde determinant defined by ##V_{n-1}(x)=\begin{vmatrix} 1 & 1 & ... & 1 & 1 \\ x_1 & x_2 & ... & x_{n-1} & x_n \\ {x_1}^2 & {x_2}^2 & ... &...

Hello, I apologize in advance for the way this post looks. I am new to this forum and I've never used LaTeX Primer. I noticed that someone has prevoiusly asked the same question, but I still do not understand how to get to the answer. Also, I tried posting an image but I could not; and this...

Homework Statement Prove the product of two Hausdorff spaces is Hausdorff Homework EquationsThe Attempt at a Solution For ##X##, ##Y## Hausdorff spaces, need to find ##s, t \in X \times Y## where neighbourhoods ##S, T## of ##s## and ##t## are disjoint. Firstly, by the definition of the...

Homework Statement ##X = \{1,2,3\}## , ##\sigma = \big\{\emptyset , \{1,2\}, \{1,2,3\} \big\}##, topology ##\{X, \sigma\}## ##Y = \{4,5\}## , ##\tau = \big\{\emptyset , \{4\}, \{4,5\} \big\}##, topology ##\{Y, \tau\}## ##Z = \{2,3\} \subset X## Find all the open sets in the subspace topology...

Homework Statement Write (13257)(23)(47512) as a product of disjoint cycles. Each bracket is a permutation of seven elements written in cycle notation. Homework EquationsThe Attempt at a Solution This isn't too hard of a problem. One easy way would be to evaluate the entire product, and then...

Homework Statement Prove the following form for an inner product in a complex space V: ##\langle u,v \rangle## ##=## ##\frac 1 4####\left| u+v\right|^2## ##-## ##\frac 1 4####\left| u-v\right|^2## ##+## ##\frac 1 4####\left| u+iv\right|^2## ##-## ##\frac 1 4####\left| u-iv\right|^2## Homework...

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...

I am given a hamiltonian for a two electron system $$\hat H_2 = \hat H_1 \otimes \mathbb {I} + \mathbb {I} \otimes \hat H_1$$ and I already know ##\hat H_1## which is my single electron Hamiltonian. Now I am applying this to my two electron system. I know very little about the tensor product...

Take seven positive integers and subtract 3 from each of them. Can the product of the resulting numbers be exactly 13 times the product of the original numbers?

Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...

Hello, I have encountered the concept of tensor product between two or more different vector spaces. I would like to get a more intuitive sense of what the final product is. Say we have two vector spaces ##V_1## of dimension 2 and ##V_2## of dimension 3. Each vector space has a basis that we...

Hi everyone, Given a vector-valued function ##\vec{A}##, how do I show that: $$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$ In other words, are the cross product and derivative commutative w/ each other? I...

Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...

In E6, the product 27 x 27 contains the (conjugate) 27. In SU(3), something similar happens with 3 x 3, which decomposes as 3 + 6. I was wondering, how usual is this? Do we have some lemmas telling when a product N x N is going to "recover" the original N, or its conjugate, inside the sum?

I am having some trouble visualising the following problem and I hope someone will be able to help me: Let (X, dx) and (Y,dy) be metric spaces and consider their product topology X x Y (T1) and the topology T2 induced by the metric d((x1,y1),(x2,y2)) = max(dx(x1,x2),dy(y1,y2)) so the maximum of...

What is the correct way to write the product rule in Newton notation (with the dots above)? It is the LHS I am abit confused with. Eg. Say you have d/dt(xy) would you just put dots above the x and y?

This problem is driving me crazy and its the last one in an assignment I've been doing for the last week please help -i(5+2i) (Sleepy)(Sadface)(Angry)

Hello Forum, Let's say we have a complete set of functions ##u_{i} (x)## that can be used to represent anyone dimensional function ##f(x)##. We then find another and different set ##v_{i} (x)## that can do the same thing, i.e. represent any function ##f(x)## via a linear superposition. I...

Hello, I try to understand the following demonstration of an author (to proove that dot product is conserved with parallel transport) : ------------------------------------------------------------------------------------------------------------------------ Demonstration : By definition, the...

Hello, I have 2 questions regarding similar issues : 1*) Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ? Is it due to the fact that angle between the tangent vector and transported vector is...

Homework Statement Let ##\prod_{n=0}^{1996} (1 + nx^{3^n}) = 1 + a_1 x^{k_1} + a_2 x^{k_2} + ... + a_m x^{k^m} ## , where ##a_1, a_2, ..., a_m ## are nonzero and ##k_1 < k_2 < ... < k_m ##. Find ##a_{1996}##. From Art and Craft of Problem Solving, originally from Turkey, 1996 Homework Equations...

Evaluate $\displaystyle \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\left(1+\frac{1}{4k^2-1}\right)$

I was trying to solve a problem involving work , as we know : w = \int_{a}^{b} \vec{f}.d\vec{s} but in my problem the path was cyrcular , so how to evaluate this kind of integral ?

$f(n)=\underbrace{111--1}\underbrace{222--2}$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n$ prove:$f(n)$ is a product of two consecutive positive integers for all $n\in N$

Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC) I tried to start with this : Let p be an arbitrary element of Ax(BΔC) then p=(x,y) such that x ∈ A and y ∈ (BΔC) x ∈ A and (y∈ B\C or y∈ C\B) (x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B) But I don't know how to continue or if I should even...

$n\in N,\,\,and \,\, a_1,a_2,a_3,-------,a_n\in Z$ $if \,\, a_+a_2+a_3+-----+a_n=a_1\times a_2\times a_3\times------\times a_n=2006$ $find \,\, min(n)$

Homework Statement Let ##Y## be some order spaced endowed with the order topology, and let ##P = \{(x,y) \in Y \times Y ~|~ x < y \}##. I would like to show that ##P## is open in ##Y \times Y##. Homework EquationsThe Attempt at a Solution Let us momentarily assume that ##Y## has neither a...

Is there a set of relationships for the wedge product of basis vectors as there are for the dot product and the cross product? i.e. e1*e1 = 1 e1*e2 = 0 e1 x e2 = e3

I'm attempting to prove that the product of two compact topological spaces is compact. My attempt at a proof runs something like this: Let ##Q## and ##R## be compact, and ##Q \times R = S##. From the product topology, any open set of ##S## has to have the form ##S_{AB} = Q_A \times R_B##...

Homework Statement T/F: If ##T: \mathbb{R}^n \rightarrow \mathbb{R}^m## is a linear transformation and ##n>m##, then the function ##\langle v , w \rangle = T(v) \cdot T(w)## is an inner product on ##\mathbb{R}^n## Homework EquationsThe Attempt at a Solution The first three axioms of the inner...

Homework Statement Let us look at a 3-dimensional potential box. Show, that the wave function in this situation can be written as the product of 3 single-argument functions. Homework Equations The 3D Schrödinger equation: \begin{equation} -\frac{\hbar^2}{2m} \left( \frac{\partial^2...

Hi! we are currently conducting our feasibility study and we're given a theme: FRUGAL INNOVATION (Doing more with less resources). we needed a product that is marketable but also have a relevance to the society (can help solve problems often faced by people e.g. traffic, pollution, crime...

I'd like to test the quality and performance of an electrical item I intend to sell. I was able to find a similar, very high quality product whose quality I'd like analyzed and compared to lower quality alternative. Who should I send these two product samples to? A third party inspection...

Let's say a company sells a toy called Tree Frog. Can the company be sued if the toy does not look exactly like a real tree frog?

The Lorentz transformation operator acting on an undotted, i.e. right-handed, spinor can be expressed as $$e^{-\frac{1}{2} \sigma \cdot \mathbf{\phi} + i\frac{1}{2} \sigma \cdot \mathbf{\theta}}.$$ There is a very cool, almost childlike, derivation of this expression in Landau Vol. 4 S. 18 I've...

Hi all, I am reading a book about fundamental quantum mechanics, in which there mentioned many time about the product state ##|a\rangle|b\rangle## of two states ##|a\rangle## and ##|b\rangle## . To my understanding, product state means to combine two small systems to get a bigger one. So I am...

I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. So in the equation ε is a symmetric matrix and δx(t) is just the difference between two points. After D.7...

Why is the Maximum Likelihood function a product? Explanations of how the Maximum Likelihood function is constructed usually just mention that events are independent and so the probability of several such events is just the product of the separate probabilities. I get the logic w.r.t...

Homework Statement (a) Let p(t) be a smooth curve in the hyperboloid S = {(x, y, z) ∈ R3 |z^2 − x^2 − y^2 = 1, z > 0}. Prove that p(t) .L p'(t) = 0, where .L is the Lorentz product. (b) Prove that any non-zero tangent vector to S has positive (Lorentz)length. (Hint: Use the Cauchy-Schwarz...

Hi! I was reading the Wikipedia article on Newton's laws of motion. I read there that when mass is a variable function of time as well as velocity, one cannot use the product rule of derivatives to expand d/dt(mv) It said that d/dt(mv)=mdv/dt+vdm/dt is WRONG I don't know why that is wrong. The...

Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product: $$\langle X, Y \rangle = tr(X Y^T),$$ and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are (1) $\langle X, X\rangle \geq 0$ and (2)...

Mod note: Reproduced contents of image with broken link: i = j x k j = k x i k = i x j Wikipedia says this about the standard basis vectors. Does this work for all (i.e, non basis) vectors? For example, if you know A = B X C does that mean C = A X B and B = C X A?