Transformation Definition and 1000 Threads

Homework Statement Write down the 2 × 2 matrix that represents the following linear transformation of the plane. Also draw the image of the (first quadrant) unit square 1. T(x, y) = (2x +6y, x + 3y). Homework Equations T(x, y) = (2x +6y, x + 3y). The Attempt at a Solution So...

Homework Statement Show that the following linear transformation matrix is a contraction mapping. \begin{bmatrix} 0.5 & 0 & -1 \\ 0 & 0.5 & 1 \\ 0 & 0 & 1 \end{bmatrix} I don't know how to make that into a matrix, but it is a 3x3 matrix. The first row is [.5 0 -1] the second row is [0...

Homework Statement Define a Function T : P3 → ℝ3 by T(p) = [p(3), p'(1), 0∫1 p(x) dx ] Show that T is a linear transformation Homework Equations From the definition of a linear transformation: f(v1 + v2) = f(v1) + f(v2) and f(cv) = cf(v) The Attempt at a Solution This is how...

Homework Statement Suppose A is an m x n matrix. (a) Let v1 ,...,vn be a basis of ℝn, and Avi = wi ε ℝm, for i = 1,...,n. Prove that the vectors v1,...,vn, w1,...,wn, serve to uniquely specify A. (b) Write down a formula for A.Homework Equations Maybe B = T-1 A S The Attempt at a Solution I...

I was thinking when I take the Lorentz formula for a location γ.(x – v.t) as an observer in S and find the ratio compared with the location for an observer within the inertial system S’ it selves: 1/γ . Δx But I made a mistake and took 1/γ. x When I use the found ratio (for derivation...

Homework Statement L: R^3 -> R^2 L(x)=(0,0)^T What is the basis, and dim of the Range? Homework Equations Rank(A)-Nullity(A)=n The Attempt at a Solution So clearly L(x)= (0,0)^T. So the basis must be the empty space and dim is zero, right? Now, going of this same logic, Say...

I want to discuss this because I afraid that the answer is no. In SR we stuck with the transformations from Poincare Group because this transformations leave invariant the exact form of the Lorentz Metric tensor. Any other transformation will change the components of the Lorentz Metric Tensor...

IR and NIR spectroscopy usually employ Fourier transformation to separate the signal into individual wavelength, UV and Vis spectroscopy normally apply gratings for light dispersion (into individual wavelength). What is the cutoff wavelength, and why is so?

Can Lorentz Transformation be applied directly to a four velocity vector? I mean let v_{α} be a four velocity vector. Is there a form of Lorentz tfm matrix such that: v^{'}_{α} = \Lambda^{β}_{α}v_{β} ?

Homework Statement We were told that it is a simple algebraic substitution to derive the t' expression from the x and x' equations for a lorentz transformation. However, I keep reaching a dead end in the algebra. Homework Equations x=B(x'+vt') x'=B(x-vt) B=1/(Sqrt(1-(v/c)^2)) B^2 = c^2/(c^2...

Hi, When I do the following transformation: $$ X_1=x_1+x_2 \\ X_2=x_2 $$ It turns out that the Jacobian ##\partial (X_1,X_2)/\partial (x_1,x_2)## is 1. But we have: $$ dx_1dx_1+dx_1dx_2=d(x_1+x_2)dx_2=dX_1dX_2=|\partial (X_1,X_2)/\partial (x_1,x_2)|dx_1dx_2=dx_1dx_2 $$ So we...

I've just started reading Arfken's book on mathematical methods for physics, and one of the very first sections is really confusing me. He is discussing the rotation of coordinates, and defining a vector as an object whose components transform in the same way as the coordinates do under a...

Hello friend, I want to know how to derive lorentz transformation. Even though i have book that derived lorentz transform,i am not able to understand. I hope you give me an easy derivation of it!

Hello friend, Can you give me an example that shows simultaneous events in one reference frame not simultaneous in other reference frame with the help of lorentz Transformation?

Homework Statement Find the matrix for the transformation which first reflects across the main diagnonal, then projects onto the line 2y+√3x=0, and then reflects about the line √3y=2x Homework Equations Reflection about the line y=x: T(x,y)=(y,x) Orthogonal projection on the x-axis...

Dear all, Poincare transformation construct a group, better to say noncompact Lie group. I want to prove this fact but I don't know how...; I mean the general characteristics- associativity, closure, identity element and inversion element. I would appreciate it if anyone could help me or...

I've been tasked with showing that a Lagrangian under a set of transformations changes by a time derivative. All has gone well, except I'm left with two remaining terms, that I am completely confident, aren't there by mistake (as the 16 terms that should be expected have all popped out with the...

A friend of mine is working with shape memory alloys and he's got one that is behaving strangely. At "low" temperatures a "fresh" solution heat treated sample will form martensite upon cooling, and austenite upon heating as expected. Heat and cool all you want and you get the transformation...

So, I'm trying (keyword trying) to learn a bit of special relativity on my own via the Stanford lectures on Youtube by Leonard Susskind, but I'm running into a problem. According to the lectures, for two different reference frames with co-ordinates marked (x, t) and (x', t'), the latter...

Homework Statement x2y'' - xy' = ln x Homework Equations The problem I'm having is what do I do with x = et or t = ln x. The Attempt at a Solution I know you have to start with x = et or t = ln x however I'm not sure what to do next...

Can anyone explain to me how I would go about transforming a Cauchy-Euler equation for an equation such as: x2y'' - xy' = ln x I know you have to start with x = et or t = ln x however I'm not sure what to do next...

Homework Statement Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian (this is a typo, they mean spherical) pψθ-space to Cartesian xyz-space is ρ2sin(ψ).Homework Equations The Attempt at a Solution Uhm, I am lost. I'm supposed to prove that when...

Question Consider the linear transformation T(x1,x2,x3)= (2*x1 -2*x2- 4*x3 ,x1+2*x2+x3) (a) Find the image of (3, -2, 2) under T. (b) Does the vector (5, 3) belong to the range of T? (c) Determine the matrix of the transformation. (d) Is the transformation T onto? Justify your answer (e) Is the...

Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? What I did was T(p)= (lamda) p = q (Lamda) p(t+1) =...

Homework Statement Let V be the linear space of all real functions Differentiable on (0,1). If f is in V define g=T(f) to mean that g(t)=tf'(t) for all t in (0,1). Prove that every real λ is an eigenvalue for T, and determine the eigenfunctions corresponding to λ.Homework Equations The Attempt...

i have a quick question, that is according to the lorentz transformation, the moving frame will have the longer time than the frame in the rest. so is that means if I'm on a moving car for my whole life, my time will greater than those who are in the rest relative to the earth?

Is there a simple way of deriving Lorentz transformation? I don't find the typical derivations in textbook so convincing, which seems to use too many intuitive postulations...

Physics books rarely make the distinction between active or passive Lorentz transformations. The usual Lorentz transformations of the spacetime coordinates in two different inertial frames seem to me to be passive transformations, because by definition passive transformations are coordinates...

Homework Statement Two spaceships A and B are launched from a point X, in opposite directions. At time t=15 minutes, spaceship A crashes. The velocity of the spaceships relative to X is 1.3x10⁸m/s. How far did the collision happen from B, as observed by astronauts on the spaceship...

I would like to get a detailed description, how the world looks for a moving observer in Special Relativity compared to the way it looks for an observer at rest. Do you know any reference, where I can find such a description? Can you maybe even tell me, where to find two pictures of the sights...

Homework Statement |\;| is a norm on \mathbb{R}^n. Define the co-norm of the linear transformation T : \mathbb{R}^n\rightarrow\mathbb{R}^n to be m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \} Prove that if T is invertible with inverse S then m(T)=\frac{1}{||S||}. Homework...

$|\;|$ is a norm on $\mathbb{R}^n$. Define the co-norm of the linear transformation $T : \mathbb{R}^n\rightarrow\mathbb{R}^n$ to be $m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \}$ Prove that if $T$ is invertible with inverse $S$ then $m(T)=\frac{1}{||S||}$. (I think probably we need...

Homework Statement See attached images below. Homework Equations For attachment "Linear 1," I've proven that it is indeed a linear transformation. My question is what does it mean when it says to show T^2=T? What exactly is the T that I am multiplying by itself? Attachment "Linear...

Homework Statement See attached image below. Homework Equations The Attempt at a Solution I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.

Homework Statement Let V = F^n for some n ≥ 1. Show that there do not exist linear maps S, T : V → V such that ST − T S = I. The Attempt at a Solution I used induction to prove that ST^n-T^nS = nT^n-1 and that S^nT-TS^n=nS^n-1, and I know I'm supposed to use that to come up with a...

Hey mobius transformation defined as f(z) = \frac{az+b}{cz+d} and ad \ne bc it is a one to one function how i can find a mobius transformation that take the real line into the unit circle I read it in the net f(z) = \frac{z - i}{z+i} and i checked it, it takes the real line into the...

Homework Statement I am trying to prove that Maxwell's laws are consistent with special relativity if one frame is moving in the x direction with another. Homework Equations In this case, I know that \frac{\partial}{\partial x'} = \gamma \frac{\partial}{\partial x} + \frac{\gamma v}{c^2}...

Homework Statement Consider the three operators defined by $$\left(S_i\right)_{jk} = -i\epsilon_{ijk}$$ in the x-y-z space and the basis vectors given in x-y-z space as $$e^{\left(1\right)} = -\frac{1}{\sqrt{2}}\left(e_x + ie_y\right), e^{\left(0\right)} = e_z, e^{\left(-1\right)} =...

There's this theorem: A linear map T: V→W is one-to-one iff Ker(T) = 0 I'm wondering if there's an analog for showing that T is onto? If so could you provide a proof? I'm thinking it has something to do with the rank(T)...

Homework Statement Let T: R3 -> M22 by T\begin{bmatrix} a \\ b \\ c \label{T} \end{bmatrix} = \begin{bmatrix} a-b & b-c \\ a+b & b+c\\ \end{bmatrix} Is this transformation one-to-one? Homework Equations The Attempt at a Solution I am not really certain...

Homework Statement Please see attached files and let me know if I am correct or not. Homework Equations The Attempt at a Solution

Hello all, I am searching for an analytic solution to an integral of the following form: I[q',k\rho\,]=\frac{1}{\pi}\int_{0}^{2\pi}e^{jq'(\phi-\phi_0)}e^{-jk\rho\sin(\phi-\phi_0)}d\phi In this equation, q' is real and k\rho is real and positive. Also, the following integral is closely...

Homework Statement Describe the possible echelon forms of the standard matrix for the linear transformation T. T: |R3 --> |R4 is one to one. The Attempt at a Solution T(x)=Ax. Right? So A must be the standard matrix. I got this: A = | £ * * | | 0 £ * | | 0 0 £ | | ? ? ? | Where £...

1. The question Let V be a vector space with the ordered basis β={v1, v2,...,vn}. Define v0=0. Then there exists a linear transformation T:V→V such that T(vj) = vj+vj-1 for j=1,2,...,n. Compute [T]β. Homework Equations [T]γβ = (aij), 1≤i≤m, 1≤j≤n (where m is dimension of γ and n is the...

Hi everyone, I am trying to solve the following problem. Is there exist a transformation matrix T, different then the block diagonal, with all blocks the same, such that the form of the matrix A=[A1 A2 ; I 0], is preserved? All blocks of A are in R^{nxn}, I is identity and 0 is zero matrix. In...

Homework Statement http://imageshack.us/a/img26/8403/homeworkprobsg28.jpg a. Use source transformations to find the voltage V0 in the circuit (green). b. Find the power developed by the 250V voltage source c. Find the power developed by the 8A current sourceHomework Equations V = IR KVL...

I'm using a book that has a loot of errors (luckly most of them are easy to recognize, like a = instead of a ≠ or viceversa, but some are way more serious), and I'm not sure if it's a new error or a thing I don't understand. Either I didn't understood all the steps of the proof or the correct...

Here is the question: Here is a link to the question: Projection and linear transformation? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement So I have this laplace transformation chart and was a bit unsure about the laplace and inverse laplace of this. The unit step function, where u(t) = 0 where t < 0, u(t) = 1 where t > 0. The laplace transformation chart that I have has two columns, the column on the...

Homework Statement Let T:P_m(\mathbb{F}) \mapsto P_{m+2}(\mathbb{F}) such that Tp(z)=z^2 p(z) . Would a suitable basis for range T be (z^2, \dots, z^{m+2}) ?