Product Definition and 1000 Threads

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?

Hey! (Mmm) Proposition The Cartesian product of two at most countable sets is countable. Proof Let $A,B$ sets both of which are at most countable. That means that there are functions: $f : \omega \overset{\text{surjective}}{\rightarrow} A, \ g : \omega \overset{\text{surjective}}{\rightarrow}...

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...

I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am...

Under what conditions are the symbols \bigoplus and \times intechangangable?

I'm interested in what people know about the application of inner product structures (usually reserved for QM) to diff equations describing classical physics, in particular non- hermitician diff operator of the Fokker-Plank equation. Thanks.

Homework Statement I am trying to figure out σt and σf for the Morlet wavelet knowing that the time bandwidth product is equal to 2.5. Any suggestion ?

The volume of a triangular prism is given by: v = ½ |a • b x c| Where b and c are two of the sides of the triangular face of the prism, and a is the length of the prism. The volume of a rectangular/parallelogram-based pyramid is given by: V = ⅓ |a • b x c| My question is, what are a, b...

Homework Statement Saw a calculation that put differentiation of power in terms of acceleration as follows: E=Fs dE/dt=Fv=P dP/dt=Fa=ma^2 It doesn't make sense to me because if power was changing, acceleration must change. Correct me if I'm wrong, but shouldn't the product rule be applied...

When we talk about the angle between two vectors while computing the cross product, which angle are we referring to exactly? According to most sources, the angle should be between 0 and π; yet according to my math book, "the angle is measured in an anticlockwise sense from a to b, if the vector...

Homework Statement If at some particular place and time the sun light is incident on the surface of the Earth along a direction defined by the unitary vector – vˆ , where vˆ =(4, 3, 5)/sqrt (50) and with a power density P, what is the total power captured by a solar panel of 1.4 m2 and with...

Hello, I'm having some difficulty with a conceptual question on a practice test I was using to study. I have the answer but not the solution unfortunately. 1. Homework Statement "For every differentiable function f = f(x,y,z) and differentiable 3-dimensional vector field F=F(x,y,z), the...

compute the product. $\left(\log_{2}\left({3}\right)\right)\cdot \left(\log_{3}\left({4}\right)\right)\cdot \left(\log_{4}\left({5}\right)\right)\cdots \left(\log_{126}\left({127}\right)\right)\cdot \left(\log_{127}\left({128}\right)\right)$ The answer to this is 7 I assume this can be done...

Homework Statement I have the operators ##D_{\beta}:V_{\beta}\rightarrow V_{\beta}## ##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}## ##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}## where each...

Hi, Assuming that A is a n x m random matrix and each of its entries are complex Gaussian with zero mean and unit-variance. Also, assume that b is a n x1 random vector and its entries are complex Gaussian with zero mean and variance=s. Then, what would be the variance of their product Ab? Any...

Hi, this is just a review exercise. Let M,N be n- and m- manifolds respectfully , so that the product manifold MxN is orientable. I want to show that both M,N are orientable. I could do some computations with product open sets of ##\mathbb R^n ## , or work with orientation double-covers...

Homework Statement I could prove a, trying b now. Homework Equations The definition of the cross prod.? The Attempt at a Solution https://www.dropbox.com/s/0sauaexkl4j2yko/proof_cross_prod.pdf?dl=0 I did not manage to get a scalar times v and a scalar times w. (No need to point this...

Homework Statement Given that \vec{V} and \vec{W} are vector operators, show that \vec{V}\times \vec{W} is also a vector operator. 2. The attempt at a solution The only way I know how to do this is by showing that the commutator with the angular momentum vector operator ( \vec{J}) is zero...

I'm asked to transform y(t) = x(t)*x(t) (where * is the convolution product) and x(t)= sinc(t)cos(2π10t) ( sinc(t)= sin(πt)/(πt) ).The attempt at a solution Clearly everything is simple if you know X(f), because y(t)=InverseFourier{ X(f)2 }. The problem is that I can't find X(f). By the way...

Homework Statement A damped harmonic oscillator is driven by a force of the form f(t)=h(t) t^2 Exp(-t), where h(t) is a Heaviside step function. The Oscillator satisfies the equation x''+2x'+4x=f(t). Use pencil-and-paper methods involving Fourier transforms and inverse transforms to find the...

Homework Statement Find vector product of C = A \times B of two vectors in oblique coord. system. Give explicit expressions of components of C in covariant and contravariant components (constructing reciprocal basis from direct basis will be useful). Homework Equations I am basically just...

Evaluate $\displaystyle \lim_{{n}\to{\infty}} \prod_{k=3}^{n}\left(1-\tan^4\dfrac{\pi}{2^k}\right)$.

I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...

Homework Statement IF G, H and G+H are invertible matrices and have the same dimensions Prove that G(G^-1 + H^-1)H(G+H)^-1 = I 3. Attempt G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1 = (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1 =2GH^-1+...

Hello, I am confused a little bit on why for a reaction with a given activation energy, one should run at high temperatures if the activation energy of the desirable product, D, is greater than the undesirable product, U. To illustrate this, I have an example with two reactions ##A + B...

Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3]. My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then, w-2x+3z=0 and 2x+y+3z=0 So, she solves the system and get the following: Subspace= {...

Hello! (Wave) Sentence: If $A,B$ are sets, there is the (unique) set, of which the elements are exactly the following: $\langle a,b\rangle: a \in A \wedge b \in B$. Proof: Remark: $\langle a,b\rangle:=\{ \{a\},\{a,b\}\}$ If $a \in A$, then $\{ a \} \subset A \rightarrow \{ a \} \in...

Hello everyone, I was wondering if the following claim is true: Let ##G_1## and ##G_2## be finite cyclic groups with generators ##g_1## and ##g_2##, respectively. The group formed by the direct product ##G_1 \times G_2## is cyclic and its generator is ##(g_1,g_2)##. I am not certain that it...

Homework Statement Hey guys, So here's the problem I'm faced with. I have to show that (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) , by acting with the quabla (\Box) operator on the following...

I have recently delved into linear algebra and multi-linear algebra. I came to learn about the concepts of linear and bi-linear maps along with bases and changes of basis, linear independence, what a subspace is and more. I then decided to move on to tensor products, when I ran into a problem...

In T. S. Blyth's book on Module Theory, the author uses a large 'times' symbol (similar to a capital X) for the Cartesian Product as seen in the text below (taken from Blyth page 58) Can someone help me with the Latex code for such a symbol?Peter

The problem: Suppose G is Abelian with two representations as the internal direct product of subgroups: G=HxK1, G=HxK2. Assume K1 is a subset of K2 and show K1=K2. My attempted solution: I took the element (e_H, k_2), where e_H is the identity element of H and k_2 is an arbitrary element in K2...

Homework Statement in the photo, the 23Na form 24Na Am i right? if i am right, the mass of 23Na should decraese , and mass of 24Na should increase. but why the solution provided is the mass of 24Na decreses and increases at the time of delta t ?? i can't understand Homework EquationsThe...