Transformations Definition and 823 Threads

1. Find out which of the transformations are linear. For those that are linear, determine whether they are isomorphisms. T(f(t)) = f'(t) + t^2 from P2 to P2 2. To be linear, T(f(t)+g(t))=T(f(t)) + T(g(t)), kT(f(t))=T(f(kt)) 3. After testing for linearity, I am thinking that the...

Hi, In component form the transformation for the following tensor can be written as F^{\mu\nu}=\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}F^{\beta\alpha} or in matrix notation, apparently as F^{'}=LFL^{T} Here L is the Lorentz transformation matrix Im happy with the component form...

Homework Statement [1] A random variable X is distributed as fX(x) = 1/9*(1+x)^2 1{-1<= x<= 2}. a) Find the density function of Y = -X^2 + X + 2. b) Find the cummulative distribution function of Y = X1{-1<=X<=1} + 1{X>=1} [2] Find the function that transforms a variable X with fX(x) =...

Lorentz transformations ("synchronising" reference frames?) Homework Statement A particle moves from (x,y,z,t) = (0 m,0 m,0 m,0 s) to (1 m,1 m,0 m,10 ns). i. What is the speed of the particle in this reference frame? ii. What is the speed of the particle in a reference frame moving...

Good evening, As an effort for trying to understand Lorentz transformations, I'm trying to use them to derive the "length contraction" result. Consider two reference frames, O (non-primed) and O' (primed), moving with respect to each other with a velocity v. Consider them to be under...

Hi, this is a question from a practice paper I have. I can't think how to do this. As far as I'm aware this has to be assumed to derive the Lorents transforms, so it must be by definition true, making the question pointless. Does anyone have any thoughts or suggestions on this? Regards, Pete

The phase flow is the one-parameter group of transformations of phase space g^t:({\bf{p}(0),{\bf{q}(0))\longmapsto({\bf{p}(t),{\bf{q}(t)) , where {\bf{p}(t) and {\bf{q}}(t) are solutions of the Hamilton's system of equations corresponding to initial condition {\bf{p}}(0) and {\bf{q}}(0)...

Homework Statement Let B be a basis of R^2 consisting of the vectors <5,2> and <1,5> and let R be the basis consisting of <2,3> and <1,2> Find a matrix P such that [x]_r=P[x]_b for all x in R^2 Homework Equations Ax=B? The Attempt at a SolutionI attempted by using Ax=B as a format to solve...

It seems that the common approach to obtain the equations for the Lorentz transformations is to guess at its form and then, by considering four separate situations, determining the values for the constants. From these equations, things like time dilation and length contraction can be worked out...

Homework Statement Determine if this is a linear transformation from R3 to R2 Homework Equations L(x) = (1 + x1, x2) The Attempt at a Solution Whenever I perform addition and scalar multiplication, I obtain this is closed under both. The book says this isn't a transformation though.

Homework Statement Derive a formula for T. T([1 1]^T)=[2 -1]^T and T([1 -1]^T)=[0 3]^T (...^T=transpose and T(...)=Linear Transformation Homework Equations T(c1v1+...+cnvn)=c1T(v1)+...+cnT(vn) The Attempt at a Solution The solutions manual's method and the method I am...

Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n: (i) P ei = λi Qei ; (ii) <P ei , ej > = δi j λi ; (iii) <Qei ...

How can I convince myself of the following statement: If x2<0, there exists L orthochronous Lorentz tranformation such that: Lx = -x My concern is this: If for example, we take xµ=(1,0,0,0), then Lx in component form is: Λµβxβ=Λµ0x0 =(Λ00, Λ10, Λ20, Λ30). By definition, if it is an...

Hi guys Ok, let's say I have a matrix given by M = \sum_{ij}M_{ij}a_i^\dagger a_j, and I wish to transform it. Now in some books I have read they write the transformation as s = \sum_{j}U_{ij}a_j, while in some notes I have read they write it as s =...

Homework Statement T: V-> V, dimV = n, satisfies the condition that T2 = T 1. Show that if v \in V \ {0} then v \in kerT or Tv is an eigenvector for eigenvalue 1. 2. Show that T is diagonalisable. Homework Equations The Attempt at a Solution I have shown in an earlier part...

Homework Statement The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3-->R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(-1,0,1) for R^3 and the basis (0,1),(1,0) for R^2. Homework Equations - The Attempt at a Solution I have no...

Homework Statement Determine if the transformation T: R^{2}\rightarrow R^{2} is linear if T(x, y)= (x+1, 2y) Homework Equations 1. T(u + v) = T(u) + T(v) 2. T(c*u) = cT(u) 3. T(0) = 0 The Attempt at a Solution I believe I have to use the above provided equations to determine...

Homework Statement Let u = (1,2), v = (3,1) and T: R^{2}\rightarrow R be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.) Homework Equations T(x)= b The Attempt at a Solution I don't really know where...

Note: in the following, parentheses denote "subscript". "G" denotes "gamma". A round, uncharged current loop is at rest in the xz plane of IRF K. The loop is centered on the Origin. Negative charge circulates around the loop, positive charge remains at rest. There is a nonzero B(y) field...

Given linear transformations S: Rn --> Rm and T: Rn --> Rm, show the following: a) S+T is a linear transformation b) cS is a linear transformation I know that since both S and T are linear transformations on their own, they satisfy the properties for being a linear transformation, which is that...

Homework Statement Two flashes of light strike at the same time, at the two orange circles on the diagram. The green train is traveling at a constant 150 kmh relative to the grey platform. The train is 1 km long. As measured by someone at point F on the grey platform, how much time passes...

Homework Statement Let V be a subset of R2 and some fixed 1-dimensional subspace of R2. F:R2->R2 by F(v) = v if v is in V, 0 otherwise Prove that F is not a linear transformation. Homework Equations The Attempt at a Solution Just wondering if i got it right, i don't want to...

Homework Statement Two spaceships approach each other, each moving with the same speed as measured by an observer on the Earth. If their relative speed is 0.70c, what is the speed of each spaceship? My current understanding of the problem. S= wrt observer on Earth S'=wrt one of the...

Homework Statement Hi all - I've been battering away at this for an hour or so, and was hoping someone else could lend a hand! Q: Show that any Mobius transformation T not equal to 1 on \mathbb{C}_{\infinity} has 1 or 2 fixed points. (Done) Show that the Mobius transformation corresponding...

If someone could explain to me how to Transform Circles ? i know how to cransform curves and such. for example there is a question asking me to transform a circle with Origin (57,8.5),r: 0.5 to a circle with orgin (57,8.5) r:6 and anther type which asks me to transform circle with orgin...

Homework Statement The origin (0,0) is in the upper left corner of the image. +x axis goes to the right while +y axis goes down. The artist draws a line from the pixel location (10,20) to the location (210,200) . She wishes to draw a second line that starts at (10,20), is 270 pixels long, and...

Hi, I was given the following problem, and i couldn't solve it yet: Give a bijection between the elements of SO(3) and the fractional linear transformations of the form \varphi_{z,w}\,(u)=\frac{zu+w}{-\bar wu+\bar z}, where u\in \mathbb C\cup \{\infty\};\, z,w\in \mathbb C. Any ideas...

Since the lorentz transformations do not change an object in the z and y directions, but it does in the x direction, is this why a ruler looks shorter in the space station example? (same everything on each station, one moving by your IFR) Also is it why the clock appears to be running slower...

Hi, I have two points on a one-dimensional Euclidean submanifold, say the x-axis. I want to assume that this subspace is kind of "cyclic". This is often accomplished with the compactification R\cup \{ \infty \} The question is: How can I compute distances (up to some constant factor)...

Hey, I have two separate questions: 1) If one is moving in a car and throws a ball straight up, say out the sun roof, the ball will have zero velocity relative to an observer in the car. Conversely, it will have the velocity of the car to a stationary observer. How does one account for drag...

I've tried several hours to understand Lorentz transformations(for space and for time)...it simply dosn't make any sense...I've posted here,on math section,because I need a better mathematical view over it... whitout this I can not understand much out of the restricted theory of relativity,thus...

dear all,we know that active transformation refers to action of changing vectors keeping the operators unchanged whereas passive transformation refers to change of operator components keeping vectors unchanged. what i cannot understand(i am just starting quantum mechanics)is in the former if we...

Homework Statement (124) If a linear transformation T : R3 -> R5 is one-to-one, then (a) Its rank is five and its nullity is two. (b) Its rank and nullity can be any pair of non-negative numbers that add up to five. (c) Its rank is three and its nullity is two. (d) Its rank is two and...

Hi there. This isn't so much a math question as it is a conceptual question. I can't seem to wrap my head around the need for coordinate transformations. *Why* do they need to be done? I think I really need a picture for this, so this might not be the right place to ask, but if you can...

Hi I have a question regarding unitary operators: If an infinitesimal operation (such as a rotation) is unitary does this guarantee that a finite transformation will also be unitary? thanks M

Homework Statement Question 3b from the following file: http://phstudy.technion.ac.il/~wn114101/hw/wn2010_hw07.pdf I know I need to find a generating function for this spacific transformation. but I don't know how to find it, I mean , how I find a spacific transformation for a spacific...

Laplace Transformations... help me please? 1. Homework Statement . Find the laplace transformations of the following: a. \sin\, {\sqrt\,{x}} b. \frac{\cos\,{\sqrt{x}}}{{\sqrt{x}}} c. \ erf\,{(t)}^\frac{1}{2}} d. \int_{t}^\infty\;\frac{\cos\,x}{x}\ e...

Homework Statement Two questions; 1. Let v1 = [-3, -4] and v2 = [-2, -3] Let T: R^2 -> R^2 be the linear transformation satisfying T(v1) = [29, -35] and T(v2) = [22, -26] Find the image of the arbitrary vector [x, y] T[x,y] = [ _ , _ ] 2. The cross product of two vectors in...

Let phi(u,v)=(u-2v,-v) is this a R^2->R^2 a linear transformation? I know that there must be two rules that must be met in order to be a linear transformation, after doing the first part, it seems that it may be linear. But I do not know how to show whether or not the second rule is...

In Chapter 1 of Blandford & Thorne: Applications of Classical Physics, section 1.7.1, "Euclidean 3-space: Orthogonal Transformations" (Version 0801.1.K), do equations 1.43 at the beginning of the section, representing respectively the expansion of the old basis vectors in the new basis, and the...

this is a problem confusing me, which is in the book named Principles of Quantum Mechanics by R. Shankar. This problem is not about quantum mechanics, but just in the chapter of Review of Classical Mechanics. (The ******** is just to avoid to be deleted). The problem is in the attachment...

Couple days ago, we get a lecture in relativity, I read quite a lot about it before so there was nothing new except one thing : our professor first started to conclude Lorentz transformation totally in a mathematical way by assuming gamma*(x-v*t) … (what I discovered that it is a known method...

Here is what the graph looks like on a graphing calculator (notice the equation at the top): http://img62.imageshack.us/img62/8898/graphingcalc.jpg Here is what my graph looks like: http://img53.imageshack.us/img53/7475/lastscanc.jpg I don't understand why the asymptote is...

Homework Statement (1) A1(2,1) --> B1(3,0) A2(0,1) --> B2(3,-2) A3(3,3) --> B3(5,1) P(4,4) The reflection, M, Maps Triangle A onto Triangle B Given that M(P) = Q, write down the coordinates of Q. C1(-3,4) C2(-3,2) C3(-5,5) The rotation R maps Triangle A onto Triangle C...

1. Question: Which of the following linear transformations T from |R^3 to |R^3 are invertible? Find The inverse if it exists. a. Reflection about a plane b. Orthogonal projection onto a plane c. Scaling by a factor of 5 d. Rotation about an axis Homework Equations The Attempt at a Solution...

Hey guys, I was wondering how you would go about proving that the image of a transformation T, im(T), is invariant? And following that, how would you prove T(W1 \bigcap W2) is invariant if T(W1) and T(W2) are both invariant. On an unrelated note, another questions asks to show that TX =...

Homework Statement Let T:U \rightarrow V be a linear transformation, and let U be finite-dimensional. Prove that if dim(U) > dim(V), then Range(T) = V is not possible. Homework Equations dim(U) = rank(T) + nullity(T) The Attempt at a Solution I almost think there must be a typo in the book...

Homework Statement Suppose T is a rotation by 30 degrees about the point 2, and S is a rotation by 45 degrees about the point 4. What is T composed with S? Can you describe this transformation geometrically? Homework Equations none The Attempt at a Solution I know T composed with S...

Hi all. I have here a reference with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. In order to calculate Noether currents, I would like to use this representation to derive formulae for the infinitesimal forms of the symmetry transformations...

Homework Statement On a supermarket parking lot, a car is pulling out and bumping into an oncoming car. The car pulls out with 0.8 m/s, while the oncoming car has a speed of 1.2 m/s. The angle between the velocities is 24 degrees, as indicated in the figure. What is the collision speed...