Transformation Definition and 1000 Threads

Homework Statement Given below are three geometrically defined linear transformations from R3 to R3. You are asked to find the standard matrices of these linear transformations, and to find the images of some points or sets of points. a) T1 reflects through the yz-plane b) T2 projects...

Algorithms like the transformation algorithm: $(x, y)$ --> $(\frac{x}{k} + p, ay + d)$ are not generally used in mathematics. Instead, we use matrices. Multiplying matrixes: you multiply a row of the first matrix by a column of the second. Use the following example: $ \begin{bmatrix}x & y...

Homework Statement Find the standard matrix of the following linear transformation: T(x1, x2, x3, x4) = (-2 x1 - 5 x2 - 4 x3 - x4, 2 x1 + 2 x2 - 5 x3 + x4) The Attempt at a Solution [x1,x2,x3,x4] [-2,2;-5,2;-4,-5;-1,1] =[-2 x1 - 5 x2 - 4 x3 - x4, 2 x1 + 2 x2 - 5 x3 + x4]...

Mod note: fixed an exponent (% --> 5) on the transformation definition.[/color] Homework Statement A is a (4x5)-matrix over R, and L_A:R^5 --> R^4 is a linear transformation defined by L_a(x)=Ax. Find the basis for the range of L_A. Homework Equations The Attempt at a Solution ##A =...

Suppose we do a constant Jacobian transformation (which is not Lorentz) of a SR (inertial) frame, by using four linear change of variables equations. This defines an apparent field with a constant metric (which is not the SR metric) in which there is relative acceleration of separation. From...

Homework Statement Consider a ##j=1, SU(2)## representation (or fundamental ##S0(3)## representation). Suppose that ##a_i, b_i## and ##c_i## (i=1,2,3) are vectors transforming under this representation i.e ##a_i' = [\rho_1 (x)]_{ij} a_j = \rho_{ij} a_j## and similarly for b and c. Consider...

Homework Statement Find the Mobius transformation which carries the points 0,1,-i to the points -1,0,\infty respectively. Find the image of the domain \{z:x<0,-x+y<t\} under this mobius transformation.Homework Equations The Attempt at a Solution Let T(z)=\frac{az+b}{cz+d}. Then...

Homework Statement T4 : R3 -> R4 is defined by T4(x1, x2, x3) = (0, x1, -3 + |x1|, x1 + x2) The Attempt at a Solution I know that T4(γ1x1 + γ2x2 + γ3x3) is a linear transformation IFF γ1.T4(x1) + γ2.T4(x2) + γ3.T4(x3) T4(λ10 + λ2x1 + λ3(-3+|x1|) = λ1.T4(0) + λ2.T4(x1) +...

T2 projects orthogonally onto the xz-plane T3 rotates clockwise through an angle of 3π/4 radians about the x axis The point (-3, -4, -3) is first mapped by T2 and then T3. what are the coordinates of the resulting point? this question is on a program call Calmaeth. My answer for this...

v=? delta what/which X(distance) over delta what/which T(time) http://en.wikipedia.org/wiki/Special_relativity

Homework Statement Say if f is a linear transformation from R2 to R3 with f(1,0) = (1,2,3) and f(0,1) = (0,-1,2). Determine f(x,y). The Attempt at a Solution I understand the theorem on linear transformation and bases but unsure as to how I should apply it in practice. Should I be...

Homework Statement We have seen that the linear transformation ##T(x_1,x_2)=(x_1,0)## on ##\mathcal{R}^2## has the matrix ##A = \left( \begin{smallmatrix} 1&0\\ 0&0 \end{smallmatrix} \right)## with respect to the standard basis. This operator satisfies ##T^2=T##. Prove that if...

Homework Statement Image Attached Homework Equations Ohm's The Attempt at a Solution Combined the two resistors in series : 250 + 550 = 800 kΩ Source Transformation (Current Source): V = 140,000(2*10^-6)= 0.28 V Combine the voltage sources : 6 - 0.28 = 5.72 V But then I...

Homework Statement Frame S' travels at speed V1 along the x-axis of frame S. Frame S'' travels at speed V2 along the x' axis of frame S'. Apply the Lorentz transformation twice to find the coordinates x'', y'', etc of any event in terms of x, y, z, t. Show that this is the same as the...

In an exercise I am asked to find the eigenvalues of a matrix A by demanding that a unitary matrix (see the attached file) diagonalizes it. I know I could just solve the eigenvalue equation but I think I am supposed to do it this rather tedious way. Now I have introduced an arbitrary unitary...

Homework Statement Ok this might be a stupid question, but: https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-frc3/t31/q77/s720x720/10001118_10202561443653973_1625797585_o.jpg Why is this the case? I think for all of this to be right, then the assumption of ##Y=u(X) \Leftrightarrow...

Hello everyone. Sorry if the question is silly, but in really need to know something. We know that The Fourier transform of time is frequency and the inverse of frequency is time. In Matlab can anyone tell me how to write it ? Because in the book Non linear fiber optics by Agrawal we found that...

Homework Statement Homework Equations The Attempt at a Solution What is M_2 supposed to be? Is A supposed to be the matrix that produces those above linear transformations?

Okay the question is to find the Fourier transform of: rect(\frac{x}{5})\otimes(\delta(x+3)-\delta(x-3)) =F^{\infty}_{\infty} \intrect(\frac{x'}{5})(\delta(x+3-x')-\delta(x-3-x')) dx' [1] - where F represents a Fourier transform. My Issue Okay I am fine doing this using the convolution...

Homework Statement A polynomial of degree two or less can be written on the form p(x) = a0 + a1x + a2x2. In standard basis {1, x, x2} the coordinates becomes p(x) = a0 + a1x + a2x2 equivalent to ##[p(x)]_s=\begin{pmatrix}a0\\ a1\\ a2 \end{pmatrix}##. Part a) If we replace x with...

Hello everyone, i am now working on a problem with three fingers robot hand to grab a cube to undergo some motion however i face some difficulties on deriving the grasp transformation matrix which help to switching the local coordinate frame at first i was given three point vectors [0 1...

The two properties every linear transformation T: V -> W has to satisfy is T(u + v) = T(u) + T(v), for u,v in V (i) T(cu) = cT(u) for u in V and scalar c (ii) I'm trying to find a transformation which satisfies (i) but doesn't satisfy (ii) [I've been able to find the opposite for what it's...

Homework Statement ##D_{ijk}## is an array with ##3^3## elements, which is not known to represent a tensor. If for every symmetric tensor represented by ##a_{jk}## $$b_i = D_{ijk}a_{jk},$$ represents a vector, what can be said about the transformation properties under rotations of the...

[SIZE="2"][I've tried asking this question on math.stackexchange.com, but haven't got any responses, so I thought I'd try here] I’m building a model in a 3D simulation program (MSC Adams) and part of that model is a triangular platform which can translate and rotate in the virtual world, as...

I am starting to learn Special and General realitivity by reading through Bernard F. Schutz's book "A First Course in General Realitivity". However I can't seem to grasp the relationship between two reference frames as compared with a Space-Time diagram. I understand the geometry of the diagrams...

Homework Statement Use resistance reduction and source transformation to find Vx in the circuit below. All resistance values are in ohms. Homework Equations The Attempt at a Solution For this problem, I know I can combine the 16 Ω resistors, but where I'm having a little trouble...

For two body decay, in CM frame, we know that the magnitude of the final particle momentum is a constant, which can be described by a delta function, ##\delta(|\vec{p^*}|-|\vec{p_0^*}|)##, ##|\vec{p_0^*}|## is a constant. When we go to lab frame (boost in z direction), what's the Lorentz...

Homework Statement A particle's movement is described by \vec{r} in the inertial system IS. Find the velocity of the particle \vec{\dot{r'}} in the system IS', which is moving with arbitrary velocity v from IS. Both inertial systems are arbitrary.Homework Equations For the position vector the...

Hi, I was wondering when you need to write the decomposition of a substance, how do you know if the number is going to be a coefficient or subscript ? Example: 2H2O ===>2H2 + 02 would be the answer But why not 2H2O ==> 2H2 + 2O Knowing that we have 2 moles of O in the beginning...

Homework Statement x(t) = 5cos(2*pi*1000*t) and g(t) = ∑ from n=-infinity to infinity delta(t-n/10000) find Fourier transform of x(t) and g(t) and the product of the two The Attempt at a Solution x(w) = 5*sqrt(pi/2) [delta(w-2000pi)+delta(w+2000pi)] g(w) = 1 so would the...

Homework Statement Starting with the Lorentz transformation for the components of the velocity, derive the transformation for the components of acceleration.Homework Equations Lorentz Transformation for position and time : ##x'={\gamma}(x - vt)## ##t'={\gamma}(t - {\frac{vx}{c^2}})## Resulting...

I've been working on a problem that I can't seem to get started on. Here is how it is posted: Metric of a space is: ds^2 = (1+2\phi^2)dt^2 - (1-2\phi)(dx^2+dy^2+dz^2), where |\phi | << 1 everywhere. Given a point (t_0 , x_0 , y_0, z_0) find a coordinate transformation to a locally...

Homework Statement let f be the linear transformation represented by the matrix M = ( -3, 2) ( 0, -2) state what effect f has on areas, and whether f changes orientation. Find the matrix that represents the inverse of f. Homework Equations N/A The Attempt at a...

http://i.imgur.com/MDigPh5.png if i have my original coordinate (white) and i am transforming this into the red coord. , could someone explain to me why y=y'cos\phi is incorrect and why y'=ycos\phi is correct?

So I've been reading Einstein's theory of relativity, and at one point when discussing the Lorentz equations' proof that light remains constant, he just states it without mathematically doing it. Probably because it wasn't the super scientific version (?) but I wanted to see how he did it, so I...

Hello, this is something basic I have hard to understand and would like to have help!:) this is a exemple from My book and I Dont understand the input! "Let $$T: P_2->P_2$$ be the linear transformation defines by $$T(P(x))=p(2x-1)$$ I Dont understand how this work $$T(1)=1, T(x)=2x-1...

Homework Statement Question as stated: In special relativity consider the following coordinate transformation between inertial frames: first make a velocity boost v_x in the x-direction, then make a velocity boost v_y in the y-direction. 1) Is this a Lorentz transformation? 2) Find the matrix...

Hi everyone, :) Here's a question I encountered recently and did partway. I need your advice on how to proceed. Question: What can be said about the Jordan normal form of a linear transformation \(f:V\rightarrow V\) where \(V\) is a vector space over \(\mathbb{C}\), if we know that...

Homework Statement Hi, I am not sure if this is the right place for my question but here goes! The stress tensor in the Si coordinate system is given below: σ’ij = {{-500, 0, 30}, {0, -400, 0}, {30, 0, 200}} MPa Calculate the stress tensor in the L coordinate system if: cos-1a33=45°, and...

Hey, I consider a diagonalized Hamiltonian: H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const with fermionic creation and annihilation...

I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...

Why ##\rho,\rho^2,\rho^3,\rho^4## are even transformation and ##\rho\sigma,\rho^2\sigma,\rho^3\sigma## are odd transformation. I'm talking about case of ##D_4## group, where ##\rho## is rotation and ##\sigma## is reflection.

Hello, I am trying to numerically evaluate a convolution integral of two functions (f*g) using Fourier transform (FT) i.e using FT(f*g) = FT(f) multiplied by FT(g) (1) I am testing for a known case first. I have taken the gaussian functions (eq. 5, 6 and 7) as given in...

Homework Statement I'm given two circles in the complex plane. |z|=1 and |z-1|=\frac{5}{2}. The goal is to find a "Linear Fractional Transformation" or Mobius Transformation that makes these two circles concentric about the origin. Homework Equations w=f(z)=\frac{az+b}{cz+d} The...

Why parity is discrete transformation? ##Px=-x## ##P\psi(x)=\psi(-x)## when ##x## is continual variable. Could you explain me difference between discrete and continual transformation?

I'm given the following transformation X=x \cos \alpha - \frac{p_y}{\beta} \sin \alpha Y=y \cos \alpha - \frac{p_x}{\beta} \sin \alpha P_X=\beta y \sin \alpha + p_x \cos \alpha P_Y=\beta x \sin \alpha + p_y \cos \alpha and I'm asked to find what type(s) of transformation it is. I'm not...

Homework Statement Let ##S = \{1, e^x, e^{-x}, e^{2x}, e^{-2x}\}## and ##B = \{1, sinh(x),cosh(x), sinh(2x), cosh(2x)\}##. S spans the vector space V, and a linear transformation T: V -> V is defined by T(y) = y'' - 3y' - 4y. (a) Find the representation matrix of T with respect to the bases S...

Homework Statement Let Q^1 = (q^1)^2, Q^2 = q^1+q^2, P_{\alpha} = P_{\alpha}\left(q,p \right), \alpha = 1,2 be a CT in two freedoms. (a) Complete the transformation by finding the most general expression for the P_{\alpha}. (b) Find a particular choice for the P_{\alpha} that will reduce the...

Hello all, I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, H = \begin{pmatrix} \xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\ -\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 &...

Hi everyone, :) Recently I encountered the following problem. Hope you can confirm whether my method is correct. My answer seems so trivial and I have doubts whether it is correct. Problem: Find the Jordan normal form of a unitary linear transformation. My Solution: Now if we take the...