Transformations Definition and 863 Threads

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focused on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices". I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4. Proposition 2.4...

The example is about the transformation between the cartesian coordinates and polar coordinates using the definition In lewis Ryder's solution, I got confused in this specific line I really can't see how is that straightforward to find?

At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following: " ... ... Each matrix A in this group determines an invertible linear transformation f_A: \mathbb{R} \to \mathbb{R} defined by f_A(x) = x A^t ... ... "I know that one may define...

Summary:: linear transformations Hello everyone, firstly sorry about my English, I'm from Brazil. Secondly I want to ask you some help in solving a question about linear transformations. Here is the question:Consider the linear transformation described by the matrix \mathsf{A} \in \Re...

This is from QFT for Gifted Amateur, chapter 14. We have a Lagrangian density: $$\mathcal{L} = (D^{\mu}\psi)^*(D_{\mu}\psi)$$ Where $$D_{\mu} = \partial_{\mu} + iq A_{\mu}(x)$$ is the covariant derivative. And a global gauge transformation$$\psi(x) \rightarrow \psi(x)e^{i\alpha(x)}$$ We are...

Hi everyone, We've just started special relativity and I'm just wondering if you'd mind clarifying something for me. The transformation is described as x'=x-vt, where x' is moving relative to x. However, in the diagram I've attached, x' is ahead of x ; so why is the transformation described...

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A). Define T:V→V,f↦df/dx. How do I prove that T is a linear transformation? (I can do this with numbers but the trig is throwing me).

Could it be that the transformations keeping the wave equation invariant have other solutions than the usual Lorentz ones ?

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 8: Differentiable Maps ... ... and am currently focused on Section 8.1 Linear Algebra ... ... I need some help in order to fully understand some remarks by Browder in Section 8.1, page...

I have to transform the first function which is f(x)=x^3 to the second function. First, I have to find each shift then combine those to make a new function equation. I've used desmos and I know that there is a horizontal shift 3 units to the right. There is a vertical shift up but I don't know...

Hi! I recently came across a quantum mechanics problem involving a change of basis to a rotating basis. As part of the solution, I wanted to transform the Hamiltonian operator into the rotating basis. Since the new basis is rotating, the basis change operator is time-dependent. This led to a...

The line element given corresponds to the metric: $$g = \begin{bmatrix}a^2t^2-c^2 & at & 0 & 0\\at & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$ Using the adjugate method: ##g^{-1}=\frac{1}{|g|}\tilde{g}## where ##\tilde{g}## is the adjugate of ##g##. This gives me...

In his book, Landau derives the Lorentz transformations using the invariance of the interval, and I have some questions about it that I would like to clarify 1. What is a parallel displacement of a coordinate system? Does it refer to moving along any axis? I don't see how any arbitrary...

T = (x+\frac{1}{\alpha}) sinh(\alpha t) X = (x+\frac{1}{\alpha}) cosh(\alpha t) - \frac{1}{\alpha} Objective is to show that ds^2 = -(1 +\alpha x)^2 dt^2 + dx^2 via finding dT and dX and inserting them into ds^2 = -dT^2 + dX^2 Incorrect attempt #1: dT= (dx+\frac{1}{\alpha})...

Hi! I am given the lagrangian: ## L = \dot q_1 \dot q_2 - \omega q_1 q_2 ## (Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case). I am...

Consider a rotating disk with the center at the origin of a stationary Cartesian coordinate system, (x, y). At t = 0, on the circumference of the disk, someone/something shoots a particle with constant velocity components Dvx, Dvy (where the D indicates the rotating disk). Also at time t=0...

Two frames measure the position of a particle as a function of time: S in terms of x and t and S', moving at constant speed v, in terms of x' and t'. The acceleration as measured in frame S is $$ \frac{d^{2}x}{dt^{2}} $$ and that measured in frame S' is $$ \frac{d^{2}x'}{dt'^{2}} $$My question...

There are several ways to show that the Lorentz transformations must be linear. What's the best/more intuitive argument in your opinion?

Hi, I'm looking for books with a really good explanation on Galilean Transformations. I find the books and/or sections where only the theory of how to convert from one to another inertial system is mentioned, but nothing with concrete examples and additional exercises. Any suggestions are...

It seems that there is a considerable number of ways of deriving the Lorentz transformations. Does anyone know how many ways are there?

In the Earth’s reference frame, a tree is at x=0km and a pole is at x=20km. A person stands at x=0 (stationary relative to the Earth), and at t=10 microseconds, this person witnesses two simultaneous lightning strikes. One of these strikes hits the tree he is standing under, and the other hits...

Given w = T (v), where T is a linear transformation and w and v are vectors, why is it that we can write any coefficient of w, such as w1 as a linear combination of the coefficients of v? i.e. w1 = av1 + bv2 + cv3 Supposably this is a consequence of the definition of linear transformations, but...

Good afternoon people. So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if: T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to...

I'm studying how derivatives and partial derivatives transform under a Galilean transformation. On this page: http://www.physics.princeton.edu/~mcdonald/examples/wave_velocity.pdf Equation (16) relies on ##\frac{\partial t'}{\partial x}=0## but ##\frac{\partial x'}{\partial t}=-v## But this...

ok this is a clip from my overleaf homework reviewing just seeing if I am going in the right direction with this their was an example to follow but it also was a very different problem much mahalo

I understand x' = λ(x - vt) but why does t' = λ(t - vx/c^2)? where does the vx/c^2 come from? and honestly I don't understand what t' is. because from what I understand is that t' is the length of time t as observed from the reference frame S'. which means t' = t*λ?

I have tried to derive the lorentz transformations but there is a part of it that requires substitution into two equations when t=0. How do I do that

I was studying linearized GR where we make the following coordinate transformation ## \tilde{x}^{a} = x^{a} + \epsilon y^{a}(x) ## This coordinate transformation is then meant to imply ## g_{ab}(x) = \tilde{g}_{ab}(x) + \epsilon \mathcal{L}_{Y} g_{ab} ## Would anyone be kind enough to explain...

In my last post I asked about the general form of the Lorentz Transformation for time. Now I am trying to understand the final form of it, and how it makes sense based on what's happening physically. The final form for t is: t = γt1 + (γv/c2/)x1 It's the second part of this equation, the...

Picture of the circuit is posted below. Apologies, the voltage source on the left should read 24 V. My question is: What is wrong with this method? [Edit: Sorry if it wasn't clear- the method in the picture yields the wrong answer] When I originally did the question, I just turned the LHS into...

By the Wigner theorem, symmetries transformations are implemented by operators ##\hat{U}## that are unitary or antiunitary. This is what is written in most books. But I have read somewhere that, to ##\hat{U}## represent a symmetrie, it's necessary that ##\hat{U}^{\dagger} \hat{H} \hat{U} =...

Photons deviate from the above energy-momentum transformations under certain circumstances while still in flat space-time, I'm wondering what set of transformations would more accurately describe them over as wide a range of circumstances as possible, still in flat space-time, I've searched and...

I am trying to understand the general form of the Lorentz Transformations before I even get into the long process of deriving that into the specific equations. In Taylor and Wheeler's, Spacetime Physics book they give this as the general form: t= Bx1 + Dt1 x= Gx1 + Ht1 In the equation for t...

Let us for a moment look a field transformations of the type $$\phi(x)\longmapsto \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right)\phi(x),$$ where ##\omega## is anti-symmetric and ##S^{\mu\nu}## satisfy the commutation relations of the Lorentz group, namely $$\left[S_{\mu \nu}, S_{\rho...

QS: Explain how the graph P(x)=3x+4 behaves under the transformation y=|P(x)| when: a) y\ge0 b) y<0 I'm not sure how to explain this in words.Thank You!

It seems that there is a difference between Galilean transformations and (the transformations of the) Galilean group, for one thing: rotations. The former is usually defined as the transformations ##\{\vec{x'} = \vec x - \vec v t, \ t' = t \}##, where ##\vec v## is the primed frame velocity...

Hi everyone! I have a problem with one thing. Let's consider the Lorentz group and the vicinity of the unit matrix. For each ##\hat{L}## from such vicinity one can prove that there exists only one matrix ##\hat{\epsilon}## such that ##\hat{L}=exp[\hat{\epsilon}]##. If we take ##\epsilon^{μν}##...

I have been searching for an answer to this for a really long time and I have not found any definitive answers as of yet. What I am trying to do is determine if and when two bodies collided between the times t0 and t1. Calculating this is much more straight forward if each body is only either...

I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ... I need some help in order to fully understand Example 6.76 ... ... Example 6.76 reads as follows: My question or...

Does General Relativity allow for transformations which are not isometries of the metric?

Hello, I'm trying to get my head around linear transformations, and there are a few things I'm not grasping too well. I'm trying to understand combinations of linear transformations, but I can't find a lot of clear information on them. As far as I can tell, any two linear transformations of the...

Hello! Do the derivatives change sign under C, P or T transformation. For example, for the photon vector field we have, under C, ##A_\mu \to -A_\mu##. Do we also get ##\partial_\mu \to -\partial_\mu ##? And what about P and T? Thank you!

One of my homework questions said "Explain how to obtain f(x)=-(3+x)^2+1 from the graph of y=x^2." I know somehow you need to move the graph right 3, reflect about the x-axis, and move up one, but I don't know how to factor and manipulate the equation to show this.

Hi there I'm studying GR and I am confused about coordinate transformations. In my understanding, if I want to study a rotating reference system this is what I do. In my inertial system the object trajectory is described by $$ x = r\cos(\theta - \omega t)\\ y = r\sin(\theta - \omega t) $$...

I have just started learning the Special Theory of Relativity. While deriving, I am facing some problems. I obviously have made some kind of mistake while using the equations... What is wrong if I don't use the time transformation equation in Event #2?

Homework Statement Homework Statement (a) Let ##V## be an ##\mathbb R##-vector space and ##j : V \rightarrow V## a linear transformation such that ##j \circ j = id_V##. Now, let ##S = \{v \in V : j(v) = v\}## and ##A = \{v \in V : j(v) = -v\}## Prove that ##S## and ##A## are subspaces and...

Homework Statement I have a mother particle at rest, which decays to a daughter particle. The daughter has mass m, momentum p and energy E and is at an angle θ1. Now I have to assume that the daughter is emitted at an angle θ2, and the mother is moving along the x-axis with velocity βc. I need...

Homework Statement Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ##...

Homework Statement \mathbb{P}^{2} is an affine plane of 2 dimensionsThe Attempt at a Solution Take for example the affine plane with z=1. Then I take a general vector v= [x,y,1] and i apply the transformation B and then the transformation A. So i get Bv=f(v) and Av=cf(v). To me this...

Are these two subjects closely related? It seems a tensor can be invariant when viewed from any **co ordinate system and The Lorentz Transformation seems to allow 2 moving co ordinate frames to agree on a space time intervals. Is there some deep connection going on? **=moving frames of...