Product Definition and 1000 Threads

(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 10.1 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 ... ... )...

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...

Homework Statement Find the gradient of \underline{\nabla}(\underline{a}\cdot\underline{r})^n where a is a constant vector, using suffix notation and chain rule. Homework Equations On the previous problem,s I found that grad(a.r)=a and grad(r)=\underline{\hat{r}} The Attempt at a Solution...

I would like to gain a more formal mathematical understanding of a construct relating to spinors. When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work...

Prove $$\sum_{n=0}^N\cos(nx)=\csc\left(\dfrac x2\right)\sin\left(\dfrac{(N+1)x}{2}\right)\cos\left(\dfrac{Nx}{2}\right)$$ I've tried working from the RHS with various identities but haven't managed to come up with anything that works. I suspect this problem involves some trigonometry that I...

Homework Statement Homework Equations The product rule formula. The Attempt at a Solution I managed to solve 45/50 product rule but I can't seem to solve these ones. Apparently you use product rule to solve these.

Let's say I have two particles A and B and I want to find the total charge parity of the system ##C_{AB}##. In what cases is it allowed to say ##C_{AB}=C_{A}.C_{B}##? I suspect that if A and B are their own antiparticles, then that is OK. Is this even the case when the system has a relative...

Homework Statement Explain in your own words why the product of eigenvalues of any diagonalisable N × N matrix A must equal the determinant of A. Homework Equations MT=M-1 The Attempt at a Solution So what I do know: the determinant measures the change in area of the unit square under the...

Homework Statement Basically, I'm looking at the property that says if the magnitude of a vector valued function is constant, then the vector function dotted with it's derivative will be zero. But I'm stuck towards the end because the proof I found online seems to skip a step that I'm not...

Homework Statement Suppose R and Q are two quantum systems with the same Hilbert space. Let |i_R \rangle and |i_Q\rangle be orthonormal basis sets for R and Q . Let A be an operator on R and B an operator on Q . Define |m\rangle := \sum_i |i_R\rangle |i_Q\rangle ...

At the risk of sounding ignorant I'd like to propose a question to someone well versed in Homological Algebra and General Relativity. I'm starting to study the tensor product functor in the context of category theory because I'm interested in possibly doing a paper on TQFT for a directed...

If the cross product in ℝ3 is defined as the area of the parallelogram determined by the constituent vectors joined at the tail, how does one go about proving this product to distribute over vector addition? I've attached a drawing showing cyan x yellow, cyan x magenta, and cyan x (magenta +...

Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused

Here is a simple question : let f(g(x)) = h(x)*g(x). I want to calculate df/dx. If I use the product rule, I get g(x)h'(x) + h(x)g'x). Now if I use the composition/chain rule, I get df/dx = df/dg * dg/dx = h(x) * g'(x) which is different. I guess my df/dg = h is wrong, but I can't see what...

Hi there, I understand that taking the dot product of two four vectors automatically applies the metric tensor to the second vector. Is there a way to take write the dot product, using vector notation, in a way which keeps the signs of all of the components positive? Thanks in advance.