Product Definition and 1000 Threads

Hi, I want to show that the Trace of the Product of a symetric Matrix (say A) and an antisymetric (B) Matrix is zero. $So\: (AB)_{ij}=\sum_{k}^{}{a}_{ik}{b}_{kj} $ $and\: Tr(AB)=\sum_{i=j}^{}(AB)_{ij}=\sum_{i}^{}\sum_{k}^{}{a}_{ik}{b}_{ki} $ $because\:A\:is\:symetric, \: {a}_{ik}=...

Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?

Homework Statement By considering A x (B x A) resolve vector B into a component parallel to a given vector A and a component perpendicular to a given vector A. Homework Equations a x (b x c) = b (a ⋅ c) - c (a ⋅ b) The Attempt at a Solution I've applied the triple product expansion and...

Hi, I am studying Sean Carroll's "Lecture notes on General Relativity". In the second chapter he identifies the volume element d^nx on an n-dimensional manifold with dx^0\wedge\ldots\wedge dx^{n-1}. He then claims that this wedge product should be interpreted as a coordinate dependent object...

Homework Statement Let f_1,f_2\colon\mathbb{R}^m\to\mathbb{R} and a cluster point P_0\in D\subset\mathbb{R}^m (domain) Prove that \lim_{P\to P_0} f_1(P)\cdot f_2(P) = \lim_{P\to P_0} f_1(P)\cdot\lim_{P\to P_0} f_2(P) Homework EquationsThe Attempt at a Solution Let \begin{cases} \lim_{P\to...

Homework Statement Need to prove that: (v⋅∇)v=(1/2)∇(v⋅v)+(∇×v)×v Homework Equations Vector triple product (a×b)×c=-(c⋅b)a+(c⋅a)b The Attempt at a Solution I know I could prove that simply by applying definitions directly to both sides. I haven't done that because that is tedious, and I...

I want to find the solution of vector X. I am using text from Alan F. Beardon Algebra and Geometry as attached. I don't know how the solution is derived for the following equation. ## x + (x × a) = b ## The second solution when ## a \times b \neq 0 ## then X cannot be b. Is it possible to...

How do you show that $$\frac{1}{z}\prod_{n=1}^\infty \frac{n}{z+n}(\frac{n+1}{n})^z$$ is meromorphic? Any hints would be helpful, I'm having trouble bounding the functions and their logarithms. This is exercise XIII.3 problem 15 in Gamelin's Complex Analysis.

Please see the attached picture. I'm not sure which hydrogen would be anti-periplanar. Any help would be appreciated. Thanks!

Hello everyone, I have thi doubt: If I have a state, say psi1, associated with the energy eigenvalue E1, the integral over a certain region gives me the probability of finding the particle in that region with the specified energy E1. Now if I put an operator between the states I obtain its mean...

In biology i have studied that in plants some secondary product examples are cuticle , lignin etc ... can you tell me why it is called secondary products ... thank you.

In a book I was reading, it says F=mv'=P' so dot producting on both sides with v F ⋅ v = mv ⋅ dv/dt = 1/2 m d(v2)/dt = d(1/2 m v^2)/dtI really don't get how v ⋅ dv/dt = 1/2 d(v2)/dt. I have seen few threads and they say it's about product rule, but they don't really explain in detail. Could...

Hi all, I am very confused on how to define the vector product or cross product in a physical sense. I know the vector product is a psuedovector, and that it is the area of a parallelogram geometrically. However, I know it used used to describe rotation in physics. As with torque, magnetism and...

Homework Statement Find the set of points of M such that: AM x BC=AM x AC (Vectors) The Attempt at a Solution [/b] AM x (BM+MC) =AMx(AM+MC) AMxBM+AMxMC=AMxAM +AM x MC Then AMxBM=0 MA X MB=0 I am new to this lesson and this is my first time i solve such a question and i had no idea...

Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though. I have the weak form as a(u,v)=l(v) where: $$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$ and $$l(v)=\int_\Omega...

$\d{}{x}{3}^{x}\ln\left({3}\right)=$ I tried the product rule but didn't get the answer😖

Hi, I have following problem of double dot product (\vec a \cdot \vec b)(\vec a^* \cdot \vec c), and I have expected rusult |a|^2(\vec b \cdot \vec c), but I don't know if it is the exactly result (I am unable to find any appropriate identity or proove it), or it is just an approximation...

Mod note: Member warned about posting with no effort. 1. Homework Statement Expand to the general case to explore how the cross product behaves under scalar multiplication k (a x b) = (ka) x b = a x (kb). The Attempt at a Solution would this be the right general case to portray the situation?

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...

Hey guys, So consider the following product of matrices: (p_{1}^{\mu}\cdot p_{1}^{\prime\nu} -(p_{1}\cdot p_{1}')\eta^{\mu\nu}+p_{1}^{\nu}p_{1}^{\prime\mu})(p_{2\mu}p_{2\nu}'-(p_{2}\cdot p_{2}')\eta_{\mu\nu}+p_{2\nu}p_{2\mu}') where eta is the Minkowski metric. I keep getting 2(p_{1}\cdot...

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

Homework Statement A and B are matrices and x is a position vector. Show that $$\sum_{v=1}^n A_{\mu v}(\sum_{\alpha = 1}^n B_{v\alpha}x_{\alpha})=\sum_{v=1}^n \sum_{\alpha = 1}^n (A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$= \sum_{\alpha = 1}^n \sum_{v=1}^n(A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$=...

I want to learn clifford and grassmannian algebras. I need to be taken from mostly a beginners point, and from a place of matrices only in general terms, and years since use. ANybody up for it? I am a software developer, so not at the bottom of any learning curve.

I've attached an image of part a of the question to this thread. My question is this (the solution to these former homework problems are posted to help us study for exam, which is why know this already): The angle between the two velocity vectors is determined to be pi/2. How? I know that dot...

Hello, I have this exercise that I can't get the right answer. I have to find derivative of g(x)= (4${x}^{2}$-2x+1)${e}^{x}$ So, what is did is g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$ My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did...

1]express j! in ∏ notation Are they just wanting something like $$j! + (j-1)! + (j-2)! +(j-3)!$$...?

I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem: I wish to calculate (\nabla \otimes \nabla) h where \nabla is defined as \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} and...

I know for two linear operators $$H_1, H_2$$ between finite dimensional spaces (matrices) we have the relations (assuming their adjoints/inverses exist): $$(H_1 H_2)^* = H_2^* H_1^*$$ and $$(H_1 H_2)^{-1} = H_2^{-1} H_1^{-1}$$ but does this extend to operators in infinite dimensions? Thanks.

I am trying to work out with Young graphs the tensor product of: \bar{3} \otimes \bar{3} The problem is that I end up with: \bar{3} \otimes \bar{3} = 15 \oplus 6 \oplus 3 \oplus 3 Is that correct? It doesn't seem correct at all (dimensionally speaking I should have taken something like...

I am trying to solve for the energy of 2 non-interacting identical particles in a 1D infinite potential well. I want to do it as much "from scratch" as possible, making sure I fully understand every step. H = -ħ2/2m * (∂2/∂x12 + ∂2/∂x22) Hψ=Eψ ∂2ψ/∂x12 + ∂2ψ/∂x22 = kψ, where k=-2mE/ħ2 I got...

Homework Statement This question has two parts, and I did the first part already I think. If B = {u1, u2, ..., un} is a basis for V, and ##v = \sum_{i=1}^n a_i u_i## and ##w = \sum_{i=1}^n b_i u_i## Show ##<v,w> = \sum_{i=1}^n a_i b_i^* = b^{*T}a## Here's how I did it: ##<v,w> =...

Homework Statement Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##: ||v - vo||^2 = ||v||^2 - ||vo||^2 Homework Equations General inner product space properties, I believe. The Attempt at a...

If I choose the positive y direction to be vertically downwards, and the positive x direction to be to the right, and take the cross product y cross x, then the direction of the resultant is out of the page (if I draw x and y as lines on paper). The magnitude is yx sin(φ), where φ is the angle...

Homework Statement Consider ##T = \delta \otimes \gamma## where ##\delta## is the ##(1,1)## Kronecker delta tensor and ##\gamma \in T_p^*(M)##. Evaluate all possible contractions of ##T##. Homework Equations Tensor productThe Attempt at a Solution ##\gamma## is therefore a ##(0,1)## tensor...

Hi I have just started looking at direct products and came across the following which i don't understand : the direct product of two spin -up vectors = | 1 > which is in a bigger vector space I don't understand how the direct product is | 1 > ? and in this case is it always a bigger vector...

The three pairs of roots $(a,\,b)$ that satisfy $a^3-3ab^2=2005$ and $b^3-3b^2a=2004$ are $(a_1,\,b_1),\,(a_2,\,b_2),\,(a_3,\,b_3)$. Evaluate $\left(\dfrac{b_3-a_3}{b_3}\right)\left(\dfrac{b_2-a_2}{b_2}\right)\left(\dfrac{b_1-a_1}{b_1}\right)$.

Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...

Homework Statement [/B] Use vectors and the dot product to prove that the midpoint of the hypotenuse of a right triangle is equidistant to all three vertices. Homework Equations [/B] I know the dot product is A⋅B = |A||B|cosΘ ... or ... A1B1 + A2B2 + A3B3 ... + AnBn I know the...

Dear all, I am trying to understand the vector triple product. ## x\times (y \times z) ## As the vector triple product of x,y and z lies in the plane ## (y \times z) ## the vector ## x\times (y \times z) ## can be written as a linear combination of the vectors ## \pm y ## & ## \pm z## In the...

Dear all, Can anyone please explain how the linear combination of non-coplanar and non-orthogonal coordinate axes representing a point x as shown below is derived. Please use the reference text attached in this post to explain to me as i will find it a bit relevant. I want to...

Hi all, I've occasionly seen people multiply a matrix by its transpose, what is the use and intuition of the product? Any help appreciated.

Homework Statement This is a general question about the equation. So, I know that the cross product requires a vector in at least 3 dimensions crossed with another. Here is the formula that I am using: uxv = My problem is the negative/positive sign orientation in front of the y element and z...

Dear all, My question is from the text of Alan F. Beardon, Algebra and Geometry concerning the scalar triple product. I have attached the text in this post. In order for the STP to be non-zero. The 3 vectors must be distinct and they are not coplanar. 2 vectors can be coplanar...

I am having trouble calculating the work done by a product gas in reversible adiabatic expansion, and in calculating the final temperature. pV gamma = constant, Cv = constant (assume), gamma = cv + nR / Cv. anyone who can help me out?

Hello! (Wave) The set $n \times m$ is equinumerous with the natural number $n \cdot m$ and thus $n \times m \sim n \cdot m$, i.e. $Card(n \times m)=n \cdot m$. Which bijective function could we pick in order to show the above? (Thinking)

Homework Statement In Sakurai's Modern Physics, the author says, "... consider an outer product acting on a ket: (1.2.32). Because of the associative axiom, we can regard this equally well as as (1.2.33), where \left<\alpha|\gamma\right> is just a number. Thus the outer product acting on a ket...

$$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$ the ans the TI gave me was $\frac{\sqrt{6}}{4}$ the derivative can by found by the product rule. but really expands the problem so not sure how the $\frac{d}{dx}$ played in this.

This seems easy but when I tried to do this, the best way I came up with is to list all entries and then do the multiplication work. Is there any better ,clearer and more simple way to do the proof?

Dear Member and Experts. We are doing a theoretical study. Kindly suggest me any Useful Product that has the probability to form with detachment of functional groups (Preferably -OH, -COOH, epoxy etc) from a surface to any reactant in environment such as H2, N2, H2O etc. or with gases such as...