Product Definition and 1000 Threads

Homework Statement Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.) PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS Homework Equations T(v)=AT*v The Attempt at a Solution I was able to prove part a. I let v=(v1,v2)...

Hey! :o Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find for the ideal $I=2\mathbb{Z}$ of $\mathbb{Z}$ the $\{m\in M\mid am=0, \forall a\in I\}$ as a product of cyclic groups. We have the following...

Hello I found a bug in my code and can't figuring out the error. I tried debugging by showing the output of each variable step by step but I can't find my error. Here is what I have and what I want to do: I have a matrix A: 0000 0101 1010 1111 And I have a matrix B: 10000 21000 30100 41100...

(Not an assigned problem...) 1. Homework Statement pg 244 of "Mathematical Methods for Physics and Engineering" by Riley and Hobson says that given the following two properties of the inner product It follows that: 2. Attempt at a solution. I think that both of these solutions are...

1. Homework Statement For part a), I know that it is a nucleophilic addition reaction of CN- to form cynohydrin, but how come there are 2N atoms in E? How does NH4+ react with CH3CHO or the cynohydrin formed? For part b), what reaction is this? Homework EquationsThe Attempt at a Solution

I recently got confused about Lie group products. Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...

Can anyone show me explicit examples of Hyperhermitian inner product?

Homework Statement A point P moves so that its distances from A(a, 0), A'(-a, 0), B(b, 0) B'(-b, 0) are related by the equation AP.PA'=BP.PB'. Show that the locus of P is a hyperbola and find the equations of its asymptotes. Homework EquationsThe Attempt at a Solution AP.PA' =...

1. Homework Statement Why is the product at anode Br2 for MgBr2? Instead of O2? Homework EquationsThe Attempt at a Solution [/B] The EΘ value for the oxidation of OH- is -0.40 V, And that of Br- is -1.07V And so, shouldn't oxidation of OH- be easier? And hence, O2 is formed at the anode?

Is there any proof for the matrixx formula of the cross product. I am asking this because I have seen many videos and they have used the matrixx formula and then proved that ||A X B|| = ||A|||B||sin(theta), khan academy also used the same method

Homework Statement Hi everyone, I am a first year physics student and we recently learned about torque. Every time I think I understand it something else comes up to confuse me - this time it is the direction. I tried looking in the forum and generally in google, but everyone only explains the...

(Scalar)·(Scalar) = Scalar (Scalar)·(Vector) = Scalar (Vector)·(Vector) = Scalar (Scalar)x(Scalar) = Not valid (Scalar)x(Vector) = Vector (Vector)x(Vector) = VectorDid I get them right, if not why? Thanks

They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices. 1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$ = $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I...

What is meant by taking the tensor product of vectors? Taking the tensor product of two tensors is straightforward, but I am currently reading a book where the author is talking about tensor product on tensors then in the next paragraph declares that tensors can then be constructed by taking...

What does the angle theta acutally means in cross product because I have seen in many places it is written that theta is the angle at which two vector on a given plane will coinside with each other so that there will be only one direction. Is it true and why they defined it in this way , I...

Why A.A = ||A||^2 , I know that from product rule we can prove this where theta =0 , I am asking this because I have seen many proves for A.B = ||A||||B||cos(theta) and to prove this they have used A.A = ||A||^2, how can they use this , this is the result of dot product formula. I havee seen...

why do we take cross product of A X B as a line normal to the plane which contains A and B. I also need a proof of A.B = |A||B|cos(theta), I have seen many proves but they have used inter product ,A.A = |A|^2, which is a result of dot product with angle = 0, we can't use this too prove...

If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...

Hi, I have this question: in the context of linear algebra, would it be correct to say that a metric is a kind of inner product?

For all real $a,\,b,\,x,\,y$ such that $ax+by=4,\\ax^2+by^2=2,\\ax^3+by^3=-1.$ Find $(2x-1)(2y-1)$.

I know that ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big) = \partial_{\mu} \phi##. Now, I need to prove this to myself. So, here goes nothing. ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu}...

A particle in a 1-D Hilbert space would have position basis states ## |x \rangle ## where ## \langle x' | x \rangle = \delta(x'-x) ## A 3-D Hilbert space for one particle might have a basis ## | x,y,z \rangle ## where ##\langle x', y', z' | x,y,z \rangle = \delta(x'-x) \delta (y-y') \delta(z-z')...

Homework Statement I am trying to determine whether $$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$ where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions. Homework Equations The defining equation of a delta function...

Hi. I'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to...

I've been working on a problem for a couple of days now and I wanted to see if anyone here had an idea whether this was already proven or where I could find some guidance. I feel this problem is connected to the multinomial theorem but the multinomial theorem is not really what I need . Perhaps...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows: I do not...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure 6.1 which is...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: About midway in the above text, just at the start...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows:https://www.physicsforums.com/attachments/5391...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ...Theorem 10.1 reads as follows: In the above text...

For any $a \in \mathbb{R}$, let $a^3$ denote $a \cdot a \cdot a$. Let $x, y \in \mathbb{R}$. 1. Prove that if $x < y$ then $x^3 < y^3$. 2. Prove that there are $c, d \in \mathbb{R}$ such that $c^3 < x < d^3$.

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ... Theorem 10.1 reads as follows:In the above...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an...

Hi all got a confusion In many books I saw , authors used a specific statement here is it a,b,c are vectors and axb is (" a cross b") In general (axb)xc ≠ ax(bxc) but if (axb)xc = ax(bxc) solving it we get bx(axc)=0 then it implies either b is parallel to (axc) or a and c are collinear...

Homework Statement e) If H ∼= Z3 × Z3 show that there are exactly 2 conjugacy classes of elements of order 2 in Aut(Z3 × Z3) = GL(2, Z3). f) Choosing an element of each conjugacy class in e), construct two semidirect products of H and K. By counting orders of elements in each such group, show...

Hello! (Wave) Does it suffice to show that the triple product is 0? If we show that $a \cdot (b \times c)=0$ we will have that $a$ is orthogonal to $b \times c$. $b \times c$ is orthogonal to both $b$ and $c$, so we will have that $a$ will be parallel to $b$ and $c$. Right? But why does...

I am reading John M. Lee's book: Introduction to Smooth Manifolds ... I am focused on Chapter 3: Tangent Vectors ... I need some help in fully understanding Lee's definition and conversation on pushforwards of F at p ... ... (see Lee's conversation/discussion posted below ... ... ) Although...

I am trying to show that if (C^ab)(A_a)(B_b) is a scalar for arbitrary vectors A_a and B_b then C^ab is a tensor. I want to take the product of the two vectors then use the quotient rule to show that C^ab must then be a tensor. This lead to the question of whether or a not the product of two...

Hello, if we consider a group G and two subgroups H,K such that HK \cong H \times K, then it is possible to prove that: G/(H\times K) \cong (G/H)/K Can we generalize the above equation to the case where HK \cong H \rtimes K is the semidirect product of H and K? Clearly, if HK is a semidirect...

In the cross product, why is vectorA*B=-(vectorB*A) How does the right-hand rule apply to this formula?

Hi all, The basis vectors are defined as 1x3 matrices, how can the result be a 3x3 matrix? How can the result of a dot product be a 3x3 matrix, I'm stumbled, how can I evaluate this? A inner product returns a scalar, and now it returns a 3x3 matrix, please help. Thanks.

We increase by 1 each of three prime numbers, not necessarily distinct. Then we form the product of these three sums. How many numbers between 1999 to 2021 can appear as such a product?

Hi, The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation Vm= (- 1/B2 ) * i *∑ ( <m,B|S|n,B> ∧ <n,B|S|m,B> ) / A2 ...[1] the textbook claims that we add the term m = n since <m|S|m> ∧ <m|S|m>...

Hi, In my QFT course, the professor writes an infinite product like this: ∏n | k n0 > 0 ∫... My question is, what does the `|' in the subscript "n | k" representing? When I see `|', I think logical OR - obviously that is not it. Normally, if it's a sum over two indices, commas separate the...