Transformations Definition and 863 Threads

Any advice on my derivation layout?

Say we have an adjoint representation (specifically in n x n matrix format) of a Lie algebra. I have noted that we can create another n x n representation using a similarity transformation on the adjoint rep. I know I haven't discovered anything new but none of my sources mention this. Is...

Alright. I was looking into a 2D rotation matrix and there are two equations: one is through the transformation of the component of p (always with respect to x,y), x,y into x',y' and the other is through the transformation of the unit vectors i,j into i',j'. In a sense 1 is passive the other is...

Homework Statement T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 Find the image of the vectors : 1. 1 2. t 3. t2 Homework Equations T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 The Attempt at a Solution I don't know how my book solves these transformations, but the answers are...

Assume the mapping T: P2 -> P2 defined by: T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 is linear.Find the matrix representation of T relative to the basis B = {1,t,t2} My book says to first compute the images of the basis vector. This is the point where I'm stuck at because I'm not...

Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?

Hi can anyone give me some hints with this question thanks A = \begin{pmatrix} 3 & -2 &1 & 0 \\ 1 & 6 & 2 & 1 \\ -3 & 0 & 7 & 1 \end{pmatrix} be a matrix for T:ℝ4→ℝ3 relative to the basis B = {v1, v2, v3, v4} and B'= {w1, w2, w3} v1 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 1 \end{pmatrix} v2 =...

Homework Statement Let V be a vector space over a field F and let L and M be two linear transformations from V to V. Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .The Attempt at a Solution I presume it's a simple question, but it's one of those where you just don't...

Why is it so that I can write: ##x'_i=A_{ij}x_j## where ##A_{ij}=\frac{\partial x'_i}{\partial x_j}##? Yes if the first expression is assumed it is clear to me why the coefficients have to be the partial derivatives, but why can we assume that we can always write it in a linear fashion in the...

(I hope this post goes in this part of the forum) Hi, I was wondering if someone could help me with the following: I have a (1+1) scalar field decomposed into two different sets of modes. One set corresponds to a Minkowski frame in (t,x) coordinates, the other to a Rinder frame in conformal...

Homework Statement Show that x = 2qa/sin(T) and p = 2qa.cos(T) is a canonical transformation into new coordinates T and momentum q. Homework EquationsThe Attempt at a Solution It looks easy, I've tried matrix/jacobi method, and symplectic method. But these two seem to be not canonical. Am I...

Suppose frame ##S^\prime## moves in the positive ##x## direction at ##v## with respect to frame ##S##, and frame ##S^"## moves in the positive ##y## direction at ##v^\prime## with respect to frame ##S^\prime##. Then, ##E^\prime_x=E_x## ##E^\prime_y=\gamma(E_y - vB_z)## ##E^\prime_z=\gamma(E_z...

1. If I know that ##H(q_i,p_i,t)## is a valid Hamiltonian for which the hamilton equations hold. Now we are given that ##Q_j(q_i,p_i)## and ##P_j(q_i,p_i)## are canonical transformations. This means that there is a function ##K(Q_j,P_j)##, the new hamiltonian, for which the Hamilton equations...

This is a screenshot from my lecture notes provided by my lecturer. He says theta=-30 and then proceeds to use a value of theta=60? Surely this is wrong?

1. Show that if we transform first in the x-direction and then in the minus x direction with the same speed (v), we end up with the original space-time coordinates. Note: For this problem you will need to apply the transformation equation twice. You will also need to apply the transformations to...

Three questions1) Let's say that N ##q_i## and ##p_i## are transformed into ##Q_k## and ##P_k##, so that: ##q_i = q_i(Q_1,Q_2,. ... , P_1,P_2, ... ) ## and ##p_i=p_i((Q_1,Q_2,. ... , P_1,P_2, ... )## We have proved that these transformations are canonical only and only if ##\forall i##...

Homework Statement Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...

When I study about the transformation of coordinates, especially while defining gradient, curl, divergence and other vector integral theorem in different co-ordinate system, a concept called metric is defined and it is said to used for transform these operators in different co-ordinates, it is...

Could anybody link me to some good examples on how to go about doing them? I honestly have no idea how to go about doing these two types of problems.

Homework Statement Give a thorough explanation as to why a linear transformation: with a standard matrix A CANNOT be one to one. Homework Equations The Attempt at a Solution I think I have figured this one out, but I was hoping somebody could confirm whether this example is sufficient...

I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean relates diffeomorphisms to active transformations. When he says this does he mean that one defines a...

Homework Statement Consider a 2x2 matrix A with A2=A. If vector w is in the image of A, what is the relationship between w and Aw? Homework Equations Linear transformation T(x)=Ax Image of a matrix is the span of its column vectors The Attempt at a Solution I know that vector w is one of the...

when we are talking about a linear transformation the argument of the function is a coordinate vector...is this true? another question...when i see a column vector...these are the coordinates of the vector with respect of a basis...is this true? for example if i see... (({{1},{3}}))^T with...

Hello, I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that is moving on a circle with a generic potential. (I used simple polar coordinates in two...

Direct experimental evidence of time slowing down for moving clocks is well-known. With the advent of atomic clocks, round trip journeys taken by these clocks on slow moving(compared to light speed) jets show time differences with clocks that have not taken the journey. But is there...

I've come up with a curious two-part question while working on a map program: What is the minimum number of points necessary in order to transform an NxN adjacency matrix into a coordinate matrix in terms of N given Euclidean space? As this question relates to map-making, where I don't...

Homework Statement Consider the lagrangian L=\delta_\mu \phi \delta^\mu \phi^* - m^2 \phi \phi^* Show that the transformation: \phi \rightarrow \phi + a \,\,\,\,\,\,\,\,\,\, \phi^* \rightarrow \phi^* + a^* is symmetry when m=0. The attempt at a solution Substituting the transformation...

Can some one explain what are the so called "large gauge transformations" and where do they play important role in physics? Explanations with less mathematical rigor will be greatly appreciated.

Well if I have a worldline given by x^{\mu}(\tau) And I want to make a Poincare transformation: x^{\mu} (\tau) \rightarrow \Lambda^{\mu}_{\nu} x^{\nu}(\tau) + a^{\mu}. I have one question,why can't a, \Lambda explicitly depend on \tau? that is to have: x^{\mu}(\tau) \rightarrow...

For the non-linear sigma action, S_G=\frac{1}{4\pi\alpha^\prime}\int d^2\sigma\sqrt{-\gamma(\sigma)}\gamma^{\mu\nu}(\sigma)G_{ij}(X)\partial_\mu(\sigma) X^i\partial_\nu X^j(\sigma), Let us consider an infinitesimal target space transformation X^\mu\to X^{\prime\mu}(X)=X+\epsilon\xi^\mu(X). The...

For example in the equation y2+x2=100, what are the transformations? what does the 100 do?

This is a follow-up to a question I asked earlier. We have the following exercise: We have two parallel mirrors, which are located at y=0 and y=l in the (x,y) plane. A photon is traveling between the mirrors, up and down along the y-axis. Consider an observer O at rest w.r.t. the mirrors...

This is the new topic we are going over in my Linear Algebra class and I am completely lost in how it works. From what I gather from the book, the lecture, and other sources amongst the internet, there are supposed to be a rule set shard with the transformation. The problem being is I cannot...

It just occurred to me that induction can be seen as a statement quite analogous to that of "a function whose derivative is 0 on an interval is constant on that interval". Suppose there is a property P about the natural numbers that we want to prove. Then let P: N -> {0, 1} be a function for...

We have two parallel mirrors, which are located at y=0 and y=l in the (x,y) plane. A photon is traveling between the mirrors, up and down along the y-axis. Consider an observer O at rest w.r.t. the mirrors. What's the time (Δt) measure by O for the photon to make a full period. Consider an...

Homework Statement A rod of length L_0 moves with a speed v along the horizontal direction. The rod makes an angle of θ_0 with respect to the x'-axis. (a) Show that the length of the rod as measured by a stationary observer is given by L=L_0\sqrt{1-\frac{v^2}{c^2}cos^2θ_0} (b)...

I am attempting to read my first book in QFT, and got stuck. A Lorentz transformation that preserves the Minkowski metric \eta_{\mu \nu} is given by x^{\mu} \rightarrow {x'}^{\mu} = {\Lambda}^\mu_\nu x^\nu . This means \eta_{\mu \nu} x^\mu x^\nu = \eta_{\mu \nu}x'^\mu x'^\nu for all x...

Homework Statement A rocket ship carrying passengers blasts off to go from New York to Los Angeles, a distance of about 5000 km. (a) How fast must the rocket ship go to have its own length shortened by 1%? (b) Ignore effects of general relativity and determine how much time the rocket...

Homework Statement Two events occur in an inertial system K as follows: Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/(2c), y2 = 0, z2 = 0 Is there a frame K' in which the two events described occur at the same place? Explain. Homework Equations Lorentz...

I have a small confusion about functions and variables. So, on doing a bit of reading, a linear transformation is a function that maps inputs from one vector space to another. So, let us take for example a simple rotation matrix. This matrix takes a point in 2D space and maps it to another...

Hello, I'm wondering what the exact definition of a local conformal transformation is, in the context of General Relativity (/Shape Dynamics) To be more precise: 1. Are local conformal transformations coordinate transformations or scalar transformations of the metric? 2. If they are...

[Mentor's note: This question was originally posted and responded to in a non-homework forum, therefore it does not have the usual homework template.] Hey, don't know how to solve this: In an inertial frame S, consider a light ray on the XY plane forming a 60 degree angle with the x-axis...

Depending on where I go to get a good understanding of the Lorentz transformations, I run into two formulas for time (t): T=T_0 * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } } and t=\left( t' + \frac{vx'}{c^2} \right) * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } } What is the...

Homework Statement Hey PF, I'm here again asking about linear transformations, ha ha. Let C={(x,y) \in \mathbb R2 | x²+y²≤1} a circle of radius 1 and consider the linear transform T:\mathbb R2→\mathbb R2 (x,y) \mapsto (\frac{5x+3y}{4},\frac{3x+5y}{4}) Find all values of a natural n for...

Hey all, I've been reading "Time Travel and Warp Drives" by Allen Everett and Thomas Roman, and the book had an interesting section on Tachyons. At one point it presented a system Leonard Parker (of the University of Wisconsin-Milwaukee) created whereby coordinate transformations for Tachyons...

Hi all, What is the difference between Lorentz transformations and yt?. That is, the Lorentz transformations for moving between two reference frames are not the same as the relativistic ones. For example considering a frame F that is stationary and an inertial frame F' with velocity v. Time...

So I made this problem up to visualize the einstein velocity transformations between inertial frames. Homework Statement I throw a frisbee due north. It goes north at a constant velocity of .7c. At the same time I throw it, a bird flies in a straight line at a constant velocity of .5c at such...

Hello, First time posting to Physics Forums. I have been thinking about rotation transformations and am a bit confused on how trig works in 3D. In 2D, convention says the positive x-axis points to the right, the positive y-axis points upward, and positive angles are measured from the...

Homework Statement Image attached. Homework Equations S-domain transformations The Attempt at a Solution Solving this using mesh analysis. I1 is straightforward : V = I1R I1 = 1.8∠75° / 2 = 0.9∠75° I2 I'm having a little trouble with. i = C dv/dt = C*V*s...

Homework Statement Prove that under an infinitesimal Lorentz transformation: x^\mu \to x^\mu+\omega^\mu_\nu x^\nu so: \phi\to\phi-\omega^\mu_\nu x^\nu\partial_\mu\phi the Lagrangian varies as: \delta \mathcal{L}=-\partial_\mu(\omega^\mu_\nu x^\nu \mathcal{L}) The Attempt at a...