Transformations Definition and 863 Threads

Diagonisability of Linear Transformations Tsquared = T Let T be a linear transformation such that T^2 = T. i. Show that if v is not 0, then either T(v) = 0 or T(v) is an eigenvector of eigenvalue 1. (easy) ii. Show that T is diagonalisable. ... Sorry, I misread the question just...

Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...

Can you please specify a reference to help me understand how QCD explains the fact that all particles observed in nature transforms under SU(3).

Homework Statement Find the equation of a sine function that has a vertical displacement 2 units down, a horizontal phase shift 60 degrees to the right, a period of 30 degrees, a reflection in the y-axis and an amplitude of 3. 2. The attempt at a solution Y = 3 sin (30x- 60) -2 I'm...

Homework Statement Event A occurs at xA = 500m. Event B occurs 5 microseconds later at xB = 1500m. With what speed must an observer move in the positive x direction so that the events occur at the same point in space in the observer's frame?Homework Equations Lorentz transformation...

How can you express the del operator after a change of variables? For example, if I want to use cylindrical coordinates for a fluids problem, what is the del operator in terms of the new coordinates? And how do you derive it for any other arbitrary coordinate transforms?

I wasn't sure if this counted as intro physics. Feel free to move if I have it in the wrong place. Homework Statement In class we learned some linear transformations where we have a stationary observer and another moving near the speed of light. Describing the reference frames: s' -> x'=u't'...

Homework Statement A3. Show that the Lorentz transformations on a spacetime 4-vector can be written as x'μ = (Lμν)*(χν) . Find the matrix L. Prove that (in matrix notation) Lτ gL = g where g is the Minkowski spacetime metric.Homework Equations Any help suggesting at least equations will be...

Homework Statement Not really a problem per se; more of an issue with some aspects of linear transformations. We've learned that a linear combination of linear transformations is defined as follows: (c_1T_1+c_2T_2)(\vec{x})=c_1T_1(\vec{x})+c_2T_2(\vec{x})\,\,\,\,\,\, \vec{x}\varepsilon...

Hello. When one is converting between coordinate systems, the Jacobian arises as a necessary consequence of the conversion. Does this occur with transformations between relativistic systems, and, if so, is this manifested through the prevalence of gamma in the transforms? Any guidance would...

1. R\circF\circR-1=S where F denotes the reflection in the x-axis where S is the reflection in the line y=x where R = R\pi/4 : R2 \rightarrow R2 3. An attempt I have found that the standard matrix for R = [cos\theta sin\theta]...

I'd like to start by mentioning that I have very little in the way of experience on the subject, so forgive me if my confusion is somewhat trivial.. My problem lies with understanding what the fundamental variables in the Lorentz Transformations actually represent. For example, it is to my...

I am struggling to understand spin transformations and have used Sakurai's method of |new basis> = U |old basis> to change basis vectors and hence should have Sz' = Udagger Sz U to transform the operator. I thought this should give Sz' = Sy in the workings (see attachment below) but it...

Homework Statement Let T \in L(V, W), where dim(V) = m and dim(W) = n. Let {v1, ..., vm} be a basis of V and {w1, ..., wn} a basis for W. Define the matrix A of T with respect to the pair of bases {vi} and {wj} to be the n-by-m matrix A = (aij), where T(v_{i}) =...

Homework Statement Let V be a real inner product space of dimension n and let Q be a linear transformation from V to V . Suppose that Q is non-singular and self-adjoint. Show that Q−1 is self-adjoint. Suppose, furthermore, that Q is positive-definite (that is, <Qv,v> > 0 for all non-zero...

I was under the impression that a source transformation doesn't change a circuit at all, which I guess is an oversimplification. If you have a series RC circuit, with a DC voltage source, after the transients have died out, all voltage will be across the capacitor, and none across the...

Homework Statement A bus travels forward at a constant speed of 24 m/s down a straight highway. the driver puts on her sunglasses, and 3.5 s later, a passanger stiing 5 m behind her drops a pen. In the frame of reference of the earth, what is the distance seprating these events? Homework...

Homework Statement Let A: E \rightarrow F be a linear transformation between vector spaces (of any dimension) and let X be a subset of F with the following property (which is only a conditional): IF X \subseteq Im(A) THEN A is surjective. ... (*) Prove that X is a generating set for...

Homework Statement Hello. I came across a question that required me to solve for invariant points between a base trig function and the function after horizontal stretch. I can't remember the exact question right now, but I'm just wondering how I would go about solving it if I didn't know...

"Handedness" of Mobius transformations Hello everyone, I've been trying to derive the SU(2) (right-handed) rotation matrix by using a projection of the sphere and Mobius transformations, but I'm having some issues which I was hoping someone here could help me out with. My apologies for the...

I've just read the statement "The Lorentz transformations have a representation on the fields" Can anyone explain the meaning of the word representation? I can't seem to get a satisfactory explanation anywhere and the notes don't go into much more detail on it.

Lets say I have Coordinate Frame's A and B. and... I have the coordinates of the 3 principle axes of B in terms of Frame A, So for a simple example, a rotation of +pi/2 about the z axis of A would yield the following mapping of the xyz axes of B in terms of Frame A: XA -> -YB YA -> XB ZA ->...

a)So I'm reading over my notes and they say that under the Lorentz transformation L, \phi \rightarrow \phi' where \phi'(x)=\phi(x') where x'^\mu = (L^{-1})^\mu{}_\nu x^\nu I don't really understand why this is true. Why is it not just \phi'(x)= L \phi(x) Clearly this fails because the LHS is...

I asked my prof why the Lorentz transformations had to be linear (which my textbook assumed when deriving them), and he mentioned some stuff about homogeneity and ended with "it's advanced, just believe". Can anyone offer a simple explanation?

Homework Statement Suppose that A is a real symmetric n × n matrix. Show that if V is a subspace of R^n and that A(V) is contained in V , then A(V perp) is contained in V perp. Homework Equations A = A_T (A is equal to its transpose) The Attempt at a Solution I have no idea...

Homework Statement [PLAIN]http://img219.imageshack.us/img219/2950/linl.jpg Homework Equations The Attempt at a Solution Is this how I do part (iii)? From (ii) I get: M^{\mathcal C}_{\mathcal C} (\phi) = \begin{bmatrix} 1 & 3 & 2 \\ 1 & -3 & 0 \\ 0 & 0 & 2 \end{bmatrix}...

Homework Statement In the old West, a marshal riding on a train traveling 35.0 m/s sees a duel between two men standing on the Earth 55.0 m apart parallel to the train. The marshal's instruments indicate that in his reference frame the two men fire simultaneously. (a) Which of the two men, the...

In a lecture on special relativity online, the form x'=x\cosh{\omega}-ct\sinh{\omega} t'=-x\sinh{\omega}+ct\cosh{\omega} is used for the lorentz transformations, where the velocity is v=\frac{c\sinh{\omega}}{\cosh{\omega}}. However, I'm wondering, couldn't you also do...

Homework Statement Prove that if V is a finite-dimensional vector space, then the space of all linear transformations on V is finite-dimensional, and find its dimension. Homework Equations The Attempt at a Solution

I'm self studying some alg topology for next semester just working through chapter 0 and 1 of hatcher really. My question is: for any universal cover p of X there are two actions of pi_1(X, x0) on the fiber p^-1(x0) given by lifting loops at x0 and given by restricting deck transformations to...

Homework Statement Hello, We were supposed to fill in the missing products or reagents (I indicated which ones on the paper) but out of 40 points, I got 13. So now I'm a bit worried. I tried to redo it, could someone look it over? (see attached jpg) The Attempt at a Solution My...

Homework Statement Write down the transformation laws under general coordinate transformations for a tensor of type (0,1) and a tensor of type (2,1) respectively The Attempt at a Solution I seem to have two transformation formulas but they could in fact just be the same thing. I'll just do...

The group of four dimensional space time symmetries may be generalised to conformal transformations x \rightarrow x' defined by the requirement dx'^2 = \Omega(x)^2 dx^2 where dx^2 = g_{\mu \nu} dx^\mu dx^\nu (recall that Lorentz invariance requires \Omega=1). For an infinitesimal...

I am having trouble with this problem: Let T:V->W be a linear transformation. Prove that T is one-to-one if and only if dimension of V = dim(RangeT). I know that in order to be a linear transformation: 1) T(vector u + vector v) = T(vector u) + T(vector v) and 2) T(c*vector u) =...

Define B( \theta, \vec{n} ) \in SL( 2 , \mathbb{C} ) by B( \theta , \vec{n}) = \cosh { \frac{1}{2} \theta} + \vec{\sigma} \cdot \vec{n} \sinh{ \frac{1}{2} \theta} where \vec{n}^2 =1 Show that this corresponds to a Lorentz boost with velocity \vec{v}=\tanh{ \theta} \vec{n}. Show that ( 1 +...

I have a metric of the form ds^2 = (1-r^2)dt^2 -\frac{1}{1-r^2}dr^2-r^2 d\theta^2 - r^2 sin^2\theta d\phi^2 A singularity exists at r=\pm 1 . By calculating R^{abcd}R_{abcd} i found out that this singularity is a coordinate singularity. I found the geodesic equations for radial photons...

Homework Statement The attachment is the problem. Homework Equations The Attempt at a Solution I understand how to go about solving the laplace transformations but I have no idea how to start with the Heaviside functions for the 5t and the 30. What I got was 5t+30U6(t) but it turned...

Homework Statement Write down 3x3 matrices A, B, C such that when the vectors in R2 are expressed in homogeneous coordinates, the product ABC first translates vectors by (-1, 2), then reflects them about the line y=-x and finally scales them by 2. using your matrix ABC, determine the image...

Hello all. I am working on a project and I am a bit stumped by something. Basically, I have 4 flat plates which can move vertically, independently from each other. I want to increase the force these plates exert upon another object. I am trying to figure out a way in which I can increase the...

hey, does anyone there know how the angular momentum (L=r x p) is transformed under Lorentz transformations?

Can someone please show me what transformation the lagrangian is invariant under in a theory with just an SU(N) gauge field (with or w/o a source term I don't think it matters). I tried to find this on my own and I got some confusing answers. Except for U(1), every source I found said that...

How can one show that if det A = 1 and the 00th component of A > = 1 then A preserves the sign of the time component of time-like vectors? thanks!

I don't quite understand the idea that (as my book says) every linear transformation with domain Rn and codomain Rm is a matrix transofrmation... I mean i get the idea of what a linear transformation is (sorta like a function) but it gives the theorem: Let T: Rn -> Rm be linear. Then there is...

I'm in a math class reading Einstein's original paper "On the Electrodynamics of Moving Bodies," from 1905. I'm stuck in Section 8, "transformation of the energy of light rays." We're basically trying to show that that Placnk's constant is Lorentz invariant- if anyone has an easy way of...

Homework Statement The Attempt at a Solution I think I first need to find T(e2)=? and T(e2)=? and then combine those into a matrix. I am having trouble starting to solve for T(e1) and T(e2) so far I have [1] = alpha [1] + beta [3] [0] [2]...

Homework Statement Consider the initial value problem: x'' + 2x' + 5x = δ(t - 1); with: x(0) = 0 and x'(0) = 0. Using Laplace transforms, solve the initial value problem for x(t). Homework Equations L[x''] = (s^2)*L[x] - s*x(0) - x'(0) L[x'] = s*L[x] - x(0) L[δ(t - 1)] = e^(-s)...

Homework Statement A light signal is sent from the origin of a system K at t = 0 to the point x = 1 m, y = 8 m, z = 13 m. a) At what time t is the signal received? b) Find ( x', y', z', t' ) for the receipt of the signal in a frame K' that is moving along the x-axis of K at a speed of 0.6c...

Homework Statement Let g(x)=3+x and T(f(x))=f'(x)g(x)+2f(x), and U(a+bx+cx2)=(a+b,c,a-b). So T:P2(R)-->P2(R) and U:P2(R)-->R3. And let B and y be the standard ordered bases for P2 and R3 respectively. Compute the matrix representation of U (denoted [U]yB) and T ([T]yB) and their...

Homework Statement Consider a two-dimensional function φ = φ(x,t) that satisfies the relativistic wave equation given by: https://adgiiq.blu.livefilestore.com/y1pe5tdBVr0r62krIiWV_PQ42r1jrzQpWKz24xRgNe138phEqCNyZJKFXhBXqqL4YCvYeAsgVQtJJwovzjL0mKiNXyd6p1zHvkx/equation.jpg?psid=1...

Show that when basic rotations are combined to find composite rota- tional transformations, if the rotation is about one of the principal axes of OXYZ (the fixed frame) the previous resultant rotation matrix is premulti- pled by the new rotational transformation, and if the rotation is about...