Transformations Definition and 823 Threads

Homework Statement The velocity of a ball in an x-y coordinate system is (10, -5) where distance is measured in metres. A second coordinate system, p-q, uses units of feet (1 ft = 0.3048 m). The p-axis is oriented at alpha = 15 degrees relative to the x-axis. The origin of the p-q system is...

Problem: Applied Partial Differential Equations (Richard Heberman) 4ed. #12.3.6 Consider the three dimensional wave equation \partial^{2}u/\partial t^2 = c^2\nabla^2 u Assume the solution is spherically symetric, so that \nabla^2 u =...

Homework Statement f(x) = 5 - g(x) Do you reflect first, then translate, or translate then reflect? Homework Equations The Attempt at a Solution So the graph would be translated up 5 units and reflected over the x-axis. Do you translate it up 5 units, then reflect? or vice versa?

Homework Statement I need to prove this formula, but I'm not sure how to prove it.[T]C = P(C<-B).[T]B.P(C<-B)-1 whereby B and C are bases in finite dimensional vector space V, and T is a linear transformation. Your help is greatly appreciated! Homework Equations T(x)=Ax [x]C=P(C<-B)[x]B...

Hi all, I've taken a two-course undergrad QM sequence and have been reading Shankar's Principles of Quantum Mechanics. There is some reference to the similarity between the Poisson bracket in Hamiltonian mechanics and the commutator in QM. E.g. \{x, p\} = 1 (PB) [x, p] = i \hbar...

Hi I am trying to do a math assignment and I am finding it really difficult. Assume you have a linear transformation from T: P3(R) --> R4 What relevance is there to applying the transformation to the basis elements of P3(R), ie: T(1), T(x), T(x^2), T(x^3)? Why is this subset special...

Homework Statement I'm not sure if this belongs in this section or in one of the physics homework sections. If it has been misposted please move it to the proper area. According to the Theory of Relativity, if an event occurs at a space-time point (x,t) according to an observer, another...

Homework Statement The sine wave sin(t) will only drive the harmonic oscillator y'' + \omega ^2 y into resonance when \omega = 1 . For what values of \omega will the half- and full-wave rectified sine waves drive the harmonic oscillator into resonance. Homework Equations The...

Can anyone explain why it's important to be able to take vectors in an x,y,z coordinate system and be able to transform them into other coordinate systems. Could not all vector considerations be grappled with in the standard x,y,z coordinate systems? How important is this ability to physicists...

I'm trying to work out how to use the Lorentz equations but so far I haven't been very successful. It would help if I had an example to let me know what I'm aiming for, so if someone would be kind enough to answer my questions about the fairly simple scenario below I would be very grateful...

Hello.The way the transformation of coordenates in Special Relativity are ussually derived presuposes linearity or try do demostrate such linearity using wrong arguments. For example some authors state that since linear and uniform motion remains linear and uniform after the transformation this...

Homework Statement Let T:P[SUB]2 -> P[SUB]2 be the linear operator by T(a[SUB]0 +a1x + a[SUB]2x = a[SUB]o + a[SUB]1 (x - 1) + a[SUB]2 (x-1)[SUP]2 Homework Equations part a ask to find the matrix [T]B - did, see below part b ask to verify matrix [T]B satisfies every vector for [T]B [X]B...

What's the problem with using dimensional analysis to derive the Lorentz transformations?

Homework Statement Let V be a finite dimensional vector space over the field F and let S and T be linear operators on V. We ask: When do there exist ordered bases B and B' for V such that [S]B = [T]B'? Prove that such bases exist only if there is an invertible linear operator U on V such that T...

I am studying invariance, and I came across this dilemma. Suppose we have a subspace with the basis <v1, v2> of the subspace (lets say U2) and we were to map v=c1v1+c2v2 and we let c2=0. Now c1T(v1)+c2T(v2)=k1c1v1+0*T(v2)= k1c1v1. I am doing a proof and need to know what the question means by...

Homework Statement In R3: T1 symmetry with respect to x -√3y = 0 & z = 0 T2 symmetry with respect to the X axis Find: The matrices for T1 and T2, T1(T2) and check that T1(T2) is a rotation around a line.Homework EquationsThe Attempt at a Solution T2 is: \begin{pmatrix} {1}&{0}&{0}&{0}\\...

Homework Statement If f(x)=\frac{2x+1}{x+2}, the equation for y=f^-1(x) is? So I switch x, x=\frac{2y+1}{y+2} The Attempt at a Solution I've tried many ways, but I must be going wrong somewhere, here's what I think to be my nearest: x(y+2)=2y+1 x(y)+2x=2y+1 x(y)+2x-2x=2y+1-2x...

http://img34.imageshack.us/img34/5391/13262160.jpg http://g.imageshack.us/img34/13262160.jpg/1/ http://img46.imageshack.us/img46/7397/62501858.jpg http://g.imageshack.us/img46/62501858.jpg/1/ http://img7.imageshack.us/img7/2651/15142727.jpg...

Homework Statement Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial. That is D(a + bx + cx2 + dx3) = b + 2cx + 3dx2: Let B be the standard basis {1; x; x2; x3}...

Homework Statement Show that (D'Alembertian)^2 is invariant under Lorentz Transformation. Homework Equations The book (E/M Griffiths) describes the D'Alembertian as: \square^2=\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} The Attempt at a Solution I don't really...

I thought this problem was pretty straightforward, but I can't seem to match the answers in the back of the book. The problem is: Find the determinant of the following linear transformation. T(v) = <1, 2, 3> x v (where the x means cross product) from the plane V given by x + 2y +...

Homework Statement Let B={b1,b2} be a basis for R2 and let T be the linear transformation R2 to R2 such that T(b1)=2b1+b2 and T(b2)=b2. Find the matrix of T relative to the basis B. The Attempt at a Solution I know that the matrix I'm looking for needs to be 2x2 and that the standard matrix...

I've recently come to the conclusion that i need to learn matrices. I read that matrices correspond to linear transformations and that every linear transformation can be represented by matrices, but what are linear transformation, and how do you represent it by a matrix?

http://img513.imageshack.us/img513/3874/85311748.th.jpg Can someone please ler me know how to go about this problem? I have made z the subject, however do not know how to advance. Thanks

Notations: L(V,W) stands for a linear transformation vector space form vector space V to W. rk(?) stands for the rank of "?". Question: Let τ,σ ∈L(V,W) , show that rk(τ + σ) ≤ rk(τ) + rk(σ). I want to know wether the way I'm thinking is right or not, or there's a better explanation...

I'm struggling to grasp what should be a trivial property of singular value decomposition. Say that I have a linear transformation T that is non-singular (i.e. T^{-1} exists) and relates matrices A and B: B = T A or A = T^{-1} B What I would like to know is how the singular values...

Assignment question: Let V = P (R) and for j >= 1 define T_j(f(x)) = f^j (x) where f^j(x) is the jth derivative of f(x). Prove that the set {T_1, T_2,..., T_n } is a linearly independent subset of L(V) for any positive integer n. I have no idea how...

The problem is T(x + yi) = x - yi Show that this is a linear transformation and find the matrix of the transformation using the following basis (1+i, 1-i) ARGH I am having trouble with the complex numbers for some reason! To show that it is linear I have to show T(x + yi...

Homework Statement T: P2 --> P2 be a linear transformation defined by T(p(x)) = xp'(x) where ' is the derivative Describe the kernal and range of T and are any of the following polynomials in the range and or in the kernal of T? 2 x2 1 - x Homework Equations power rule (for...

Homework Statement The matrix \left[ \begin{array}{ccc} 0 &1 &0 \\ 0 &0 &1 \\ 1 &0 &0 \end{array} \right] represents a rotation. (a) Find the equation of the axis of this rotation. (b) What is the angle of the rotation? Homework Equations \left[ \begin{array}{ccc} 1 &0 &0...

Do canonical transformations simply transform the coordinates of a particular system, leaving the physics unchanged? or can they transform between physically different systems? I haven't seen any evidence which shows that they keep the physics the same, but I don't see their usefulness otherwise.

I'm having some difficulty understanding how to perform linear transformations on matrices. I understand the definition but not how to perform the operations. I'm going to give a few examples from my book: Suppose that T: R^2 \longrightarrow R^2 is a linear transformation such that...

In the story below, where would you see possible alternatives, or where would you see a problem? (01) Let us consider a set of physicists {P0, P1, P2, P3, ...} each at rest in their own inertial frames. (02) Let us elect one of them (P0) as the boss to manage an experiment. (03) Let us...

Can anyone explain the above-i've read about in books, internet sites and still do not understand what its doing or the maths. Thanks

Can anyone explain the above-i've read about in books, internet sites and still do not understand what its doing or the maths. Thanks

1. A cosmic-ray proton streaks through the lab with velocity 0.85c at an angle of 50o with the +x direction (in the xy plane of the lab). Compute the magnitude and direction of the proton’s velocity when viewed from frame S’ moving with β=0.72 in the +x direction. 2. Ux'= Ux-v/1-vUx/c^2...

1. Using the Lorentz Transformations, show that the quantity px - Et is invariant, where p and E are the momentum and energy, respectively, of an object at position x at time t. 2. px - Et 3. I needed help on starting the problem. Where should I begin?

Question and solution http://img141.imageshack.us/img141/4881/74319855am1.th.jpg http://img141.imageshack.us/img141/1295/58915812pu1.th.jpg Can someone please explain the solution - why is the right hand matrix t(k-4) t(1+k)? Thanks a lot in advance

1. Suppose V,W are vector spaces over a field F and that T: V ---> W is a linear transformation. Show that for any v belonging to V that T(-v) = -T(v) 2. -T(v) denotes the additive inverse of T(v) 3. I think I'm really overcomplicating it =/ But i have 0v = T( v - v ) = T(v) +...

Hey, not strictly homework but this is probably the best place for it, I wonder if you guys can help me out with a past paper question I've been pondering: Two events occur at the same place in an inertial reference frame S, but are separated in time by 3 seconds. In a different inertial frame...

I'm reviewing for exams and don't understand when to use which Lorentz velocity equation to use. one goes v'=(v-u)/(1-vu/c^2) and the second v=(v'+u)/(1+v'u/c^2)

A = \left[\begin{array}{ccccc} 1 & -1 \\ 2 & 5 \\ 3 & 4 \end{array}\right] Let T_{A}: R^2 \rightarrow R^3 be the matrix transformation that maps a 2 \times 1 column vector x in R2 into the 3 \times 1 column vector Ax in R3. The relationship can be expressed as TA(x) = Ax Find a vector...

Hi, I would like to transform a vector from Spherical to cartesian coordinate system. But the question is probably not that straight forward. :( I have a vector say E = E_r~\hat{r}+E_{\theta}~\hat{\theta}+E_{\phi}~\hat{\phi}. But I know only the cartesian coordinate from where it...

Prtesent the Lorentz transformations as dx=g(dx'+Vdt') dt=g(dx-Vdt) In my oppinion dx and dx' represent proper lengths measured in I and in I', dt and dt' representing coordinate time intervals. Do you aggree. Happy new year to all participamts on the Forum

As I try to understand GR, I find coordinate transformations just about everywhere. My question is simply: What is the reason coordinate transformations play such an important role in GR? Thanks.

Homework Statement Find a basis for V. Let W be a vector space of dimension 4. Let beta = {x1, x2, x3, x4 } be an ordered basis for W. Let V = {T in L(W) | T(x1) + T(x2) = T(x4) } Homework Equations L(W) is the set of linear transformations from W to W The Attempt at a Solution...

Homework Statement Determine whether the subset W of the vector space V is a subspace of V. Let V = L(Q4) (the set of linear transformations from rational numbers with 4 coordinates to rational numbers with 4 coordinates). Let W = { T in V = L(Q4) | { (1,0,1,0) , (0,1,0,-1) } is contained in...

Homework Statement Hi all, we just started doing linear transformations in class and I still don't fully understand them. Here's one question I've been stuck on: Let P2(x,y) be the vector space of polynomials in the variables x and y of degree at most 2. Recall the monomial basis for this...

Separating a fraction I don't remember what this method is named in English, but I want to write the fraction \frac{1}{(s^2 + 1)(s-3)(s+2)} in the form \frac{A}{s^2 + 1} + \frac{B}{s-3} + \frac{C}{s+2} I multiply A with (s-3)(s+2), B with (s^2 + 1)(s+2) and C with (s^2 + 1)(s-3)...

I was reading a statistics book, and part of the problem reduces to the calculus problem of doing the following: 1) Let u=x/y, v=y, with domain 0<x<y<1[/color], how to find the ranges of u and v after the transformation? 2) Let u=x/(x+y), v=x+y with domain x>1, y>1[/color], what values...