Transformation Definition and 1000 Threads

Homework Statement You are given that T is a linear transformation from R^3 to P2, that T((1,1,-1)) =X, and that T((1,0,-1))=X^2+7X-1. Find T(0,-5,0) or explain why it cannot be determined form the given information. Homework Equations None The Attempt at a Solution There is only X...

So I was told by my teacher today that I am doing the bogoliubov transformation wrong because I'm supposed to have 4 different vectors, i.e Instead of Ak and Bk A1k A2k and B1k B2k I'm wondering how this makes sense. I've seen papers with 2 different vectors and their complex...

I'm trying to find the set $\mathscr{F}$ of all linear fractional transformations (l.f.t.) of the unit disc D in itself which map 1 in 1, -1 in -1 and i in -i. By l.f.t. i mean a function$$f(z)=\frac{az+b}{cz+d}$$with $a,b,c,d\in\mathbb C$, $ad-bc\neq0$.I know that this kind of maps sends lines...

Consider the affine transformation \(f(P)=\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}P+\begin{bmatrix}5\\6\end{bmatrix}\). Find the image of \(ax+by+c=0\) under \(f\). My answer is \(\left(a-\frac{b}{2}\right)y+\left(\frac{3b}{2} -2a\right)x+4a-\frac{9b}{2}+c=0\).

I have 2 coordinate systems which move along ##x,x'## axis. I have derived a Lorentz transformation for an ##x## component of momentum, which is one part of an 4-momentum vector ##p_\mu##. This is my derivation: \scriptsize \begin{split} p_x &= mv_x \gamma(v_x)\\ p_x &= \frac{m...

Using linear transformation reflection to find rotation Homework Statement Let T1 be the reflection about the line −4x−1y=0 and T2 be the reflection about the line 4x−5y=0 in the euclidean plane. The standard matrix of T1 \circ T2 is what? Thus T1 \circ T2 is a counterclockwise rotation...

Suppose I have a transformation (x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2)) and I wish to find \partial x'_1\over \partial x'_2 how do I do it? If it is difficult to find a general expression for this, what if we suppose f,g are simply linear combinations of x_1,x_2 so something like ax_1+bx_2 where...

Hi, if we suppose x and y are two elements of some vector space V (say ℝn), and if we consider a linear function f:V→V', we know that the inner product of the transformed vectors is given by: \left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} ...

Homework Statement So there's a linear transformation T: ℝ3 → ℝ4, standard matrix A that satisfies det(A e1) = 5, det (A e2) = 4, det (A e3) = 5 and det (A e4) = 5 If S is the unit sphere, find the 3-dimensional volume of T(S). Homework Equations Volume of sphere = 4/3 * pi * r^3...

Let T:R^2 -> R^2 be the linear transformation that projects an R^2 vector (x,y) orthogonally onto (-2,4). Find the standard matrix for T. I understand how to find a standard transformation matrix, I just don't really know what it's asking for. Is the transformation just (x-2, y+4)? Any...

Hi there, The Law of Cosines can be stated as a^2 = b^2 + c^2 - 2bccos(A) where a,b, and c are the sides of a triangle, and A is the angle opposite the side a. I have a function, f(b,c,A), with an associated set of partial derivatives (\frac{∂f}{∂c})_{b,A} etc. What I want to do is to...

Homework Statement Hi! i want to ask somebody who are studying quantum mechanics about the definition of regular transformation. I guess there might be people who are not familiar with the notion. So, i'd like to let you know which book I'm referring to; "principles of quantum mechanics" ...

Homework Statement I'm trying to get to grips with Godel's 1949 Paper on Closed Time-like Curves (CTCs). Currently I'm trying to confirm his transformation to cylindrical coordinates using maple but seem to keep getting the wrong answer. Homework Equations The line element in cartesian...

Hi all, I have a linear algebra question relating actually to control systems (applied differential equations) for the linear system {\dot{\vec{{x}}} = {\bf{A}}{\vec{{x}}} + {\bf{B}}}{\vec{{u}}}\\ \\ A \in \mathbb{R}^{ nxn }\\ B \in \mathbb{R}^{ nx1 }\\ In class, we formed a...

My question is in the paint document. And I think I know the answer to my question. I asked why can't I let v = 1 then my first first region transformation would the line y = b between -a≤x≤a. The reason I think I can't do this is because the end point v = 1 is a point and not a line...

Hi We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel, U: the 2x2 symmetric matrices (ab) (bc) A basis for U is (10)(01)(00) (01)(10)(01)I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it...

Consider the time-reversal Lorentz transformation given by the 4x4 matrix: \Lambda_T = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 &1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &1 \end{pmatrix}. In my relativistic quantum mechanics lecture, we discussed how the electromagnetic 4-potential...

okay, so I'm at the most elementary stage of learning limits and there are things which leave me baffled at times, namely two. 1. lim (x -> a) f(x) = lim (x+k -> a+k) f(x) how? the physical reason behind this? 2. the theorem to evaluate limits of the form --- 1^infinity if f(x)=g(x)=0...

Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...

I'm just getting started on relativity. I watched this a couple of day ago - But I didn't like the way Lorentz Transformation was derived (the assumption about the nature of the final transformations, to be more specific). I tried reading Einstein's original paper for a better derivation but...

The Title pretty much says it all. I'm trying to learn how to solve the Inverse Laplace Transformation of Arctan(s/2). An equation of this sort was not explicitly covered in class and I'm having difficulty figuring where to start to solve it. If anyone could give me a general idea that would...

Homework Statement A Particle moves with uniform speed V'y = Δy'/Δt' along the y'-axis of the rocket frame. Transform Δy' and Δt' to laboratory displacements Δx, Δy, and Δt using the Lorentz transformation equations. Show that the x-component and the y-component of the velocity of this...

Homework Statement Here is the problem with my attempt at the solution: The magnitude of my answers are correct, HOWEVER I am getting the wrong signs. For the force balance in the x direction I get a negative P but for the force balance in the y direction I get a positive P. Does anyone...

Homework Statement Define L: R(mxm) to R(nxn). If L(A)=L(B), prove or disprove that det(A)=det(B). Homework Equations The Attempt at a Solution I think I can prove that this is true. L(A)=L(B) means that L(A)-L(B)=L(A-B)=0. Now let C be the matrix representation of L. We...

I am trying to build a rotational transformation matrix both for counterclockwise and clockwise angles. The first matrix in the picture is for counterclockwise angles and the second one for clockwise angles. The first matrix I built corresponds to the one given in my linear algebra book so it...

Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds: (i) the equation T(x)=b has a solution for all vectors b in V. (ii) Nullity of T>0

I am currently looking a bit into special relativity. Consider the matrix \Lambda=\left( \begin{array}{cc} \gamma & -\gamma \beta c \\ -\gamma \beta c & \gamma \end{array} \right) where \beta=\frac{v}{c},\quad \gamma=\frac{1}{\sqrt{1-\beta^2}} and c is the speed of light. Then, an observer...

Hello all ! Homework Statement I have the following problem. I have to calculate the DTFT of this : x(n)=u(n)-u(n-4). Homework Equations Fourier Transformations The Attempt at a Solution So far , from what I have studied I have understood, that a DTFT , is actually many...

The problem is attached. The problem is "find a basis for the range of the linear transformation T." p(x) are polynomials of at most degree 3. R(T)={p''+p'+p(0) of atmost degree 2} This is pretty much as far as I got. I'm not sure how to do the rest. I'm thinking of picking a...

T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2. T (p(x)) = p(0) Find a basis for nullspace of linear transformation T.The answer is {x, x^2} I want to make sure I'm interpreting this correctly. It...

I attached the problem, idk if it's really easy or If I'm doing it all wrong. Since T is a linear transformation T(u+v)=T(u)+T(v)=w+0=w?

Homework Statement Ok so a Kaon (m = 500MeV) is accelerated from rest along the z-axis to a final energy of 5GeV, I need to find two factors of a lorrentz transformation β and γ and write a four vector for this. Homework Equations β=p/E γ=E/m The Attempt at a Solution I have...

This thread is spawned from an earlier one https://www.physicsforums.com/showthread.php?t=647147&page=7 For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\...

Homework Statement A more efficient algorithm to calculate Fibonacci numbers applies the simultaneous transformation: T(a; b) = (a+b; a) repeatedly with a = 1 and b = 0 as initial values. What Fibonacci numbers result from T^k(1; 0)? Justify your answer (e.g., as proof by induction in...

Concerning Rapidity, if tanh(Fi) = v/c, can it be concluded in general that the relative angle of two frames in combination with Lorentz Transformation is tan(theta) = tanh(Fi) = v/c, where theta is the relative angle?

Homework Statement Let V be an inner product space. For v ∈ V fixed, show that T(u) =< v, u > is a linear operator on V . Homework Equations The Attempt at a Solution First to show it is a linear operator, you show that T(u+g)=T(u)+T(g) and T(ku)=kT(u) So, T(u+g)=<v...

Homework Statement Suppose T: V -> W is linear. Prove that T(0) = 0 The Attempt at a Solution T(v) = Av T(0) = A(0) = 0 Is that right?

Homework Statement Given that the derivative θμ transforms as a covariant vector ,show that θμ transforms as a contravariant vector. Homework Equations Please look the attachement The Attempt at a Solution Does anyone know how i should go to prove it ?Is it just a trivial...

Is the answer correct?

Homework Statement Show that the following is a Lorentz Transform: \Lambda _{j}^{i}=\delta _{j}^{i}+v^iv_j\frac{\gamma -1}{v^2} \Lambda _{j}^{0}=\gamma v_j , \Lambda _{0}^{0}=\gamma , \Lambda _{0}^{i}=\gamma v^i where v^2 =\vec{v}\cdot \vec{v}, and \delta _{j}^{i} is the Kronecker Delta...

There's a part in my book that I don't understand. I have attached the part and it is basically about how to transform from a set of conjugate variables (q,p) to another (Q,P) while preserving the hamilton equations of motion. I don't understand what he means by q,Q being separately independent...

I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure. Of course, because of the 2-dimensional case they are local conformally flat. I am looking for a relation between...

Hello. I don't understand one transformation that is made on page 25 of this paper: http://www.atm.ox.ac.uk/user/read/mechanics/LA-notes.pdf It is the second equation from the top, ont the one marked as '2', but just the second one. dx/dt*(\delta x*dx/dt)=1/2\delta(dx/dt)^2 Why...

Suppose that T1: V → V and T2: V → V are linear operators and {v1, . . . , vn} is a basis for V . If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show that T1(v) = T2(v) for all v in V . I don't understand this question. They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n...

I attached the problem. I'm not sure if I'm misinterpreting the question, but this problem seems really easy, which is usually not the case with my class. for part a) isn't that just the coefficient matrix of the right hand side? This makes A: 1 -2 3 1 0 2 for part b) T(e1)=T[1...

Find under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are: Q = ap/x , P=bx2 And apply the transformation to the harmonic oscillator. I did the first part and found a = -1/2b I am unsure about the next part tho: We have the...

Hi all, I'm a part III student and taking the QFT course. The following seems "trivial" but when I went and asked the lecturer, the comment was that they too hate such nitty gritty details! The problem is page 12 of Tong's notes: http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf All...

The problem statement has been attached. To show that T : V →R is a linear function It must satisfy 2 conditions: 1) T(cv) = cT(v) where c is a constant and 2) T(u+v) = T(u)+T(v) For condition 1) T(cv)=∫cvdx from 0 to 1 (I don't know how to put limits into the integral...

Homework Statement As the title suggests, I need help finding resources that clearly shows the step by step process of the derivation of the rest or invariant mass using the Lorentz transformation. Homework Equations Energy-momentum relation The Attempt at a Solution Not looking...

Let T: P2(ℝ) --> P2(ℝ) be defined by T(p(x)) = p( x-1) a) Find the matrix of T with respect to the standard basis of P2(ℝ) Question: So I know how to do this for the most part, I'm just having a problem in terms of the constant 1 from the standard basis of {1, x , x2 from P2(ℝ)...