Transformation Definition and 1000 Threads

I know De-Morgan's law that $$ -(p∧q) = -p∨-q $$ Also $$ -(p∨q) = -p∧-q $$ But for material implication and bi conditional operations there are also some transformation. What is the law or proof for it? Like $$ p⇒q = -p∨q $$ $$ p ↔q = (p∧q) ∨ (-p∧-q) $$ There may be other properties also that I...

Hello people, I have a question regarding the x' component in the Lorentz/Galilean transformation. So from what i understand is that there are 2 coordinate systems used in the transformations. One is used as a reference point and one is used for moving away from this point. The moving away in...

Hi all, I have the definition of a linear transformation in terms of a transformation matrix. So the mapping is a function $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$, where $f(\textbf{x})=A\textbf{x}$ and $A$ is a $n\times m$ matrix. I'm looking for a similar definition for a transformation that...

Homework Statement Find the domain, target space, image, rank and nullity of the linear transformation T(A)=Av, where v= (1, 2) and A is any 2×2matrix. Homework Equations The Attempt at a Solution I believe I know the domain (R2x2 vector space) and target space (R2), but I am not sure how to...

I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part. Do I need to use pictures? This is Exercise $23$ in Section $3.3$ of Conway's *Functions of One Complex...

So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##* I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2), where ##r*=r+2m In(\frac{r}{2m}-1)## and to the coordinate system ##v,r,\phi, \theta ##, where...

I'm trying to figure out how the element of solid angle transforms under a transformation between two inertial frames moving with velocity v w.r.t. each other under an arbitrary direction. But I should say I disappointed myself! Anyway, some books which contain a brief discussion on this(which...

If two 5-year survival probabilities are p1=.55 and p2=.41 the ratio is .55/.41 = 1.34 but since probabilities are in [0, 1] should I take the log first? Which is the more appropriate way to interpret the ratio? the ratio of logs is Log(.55)/log(.41) = .671 Which is less than one although...

Linear transformation D:Psub2 to Psub2 defined by D( Asub0 + Asub1x + Asub2x^2) = Asub1 + 2Asub2x Find the matrix of this linear transformation with respect to the ordered bases C to C, where C= { 1-x , 1+ x, x^2 } I know that D stands for differentiating . D prime is Asub1 + 2Asub2x I...

Homework Statement Find a linear transformation w = f(z) such that it maps the disk Δ(2) onto the right half-plane {w | Re(w) > 0} satisfying f(0) = 1 and arg f'(0) = π/2 Homework Equations w = f(z) = \frac{az+b}{cz+d} z = f^{-1}(w) = \frac{dw-b}{-cw+a} The Attempt at a Solution [/B]...

This is one of those "existential doubts" that most likely have a trivial solution which I can't see. Veltman says in the Diagrammatica book: Although the reasoning makes perfect sense for a Hilbert space spanned by momentum states, intuitively it doesn't make sense to me, because a...

Homework Statement If T is linear, show that it is linear by finding a standard matrix A for T so that: Also show that this equation holds for the matrix you have found. If T is not linear, prove that T is not linear by showing that it does not fit the definition of a linear transformation...

Hello Guyz I've got a little Question For Seniors I hope You answer it Briefly , I know that whenever i plug in my USB Drive in my Loudspeaker or plug Speaker itself in my PC and play anything like a song an Electrical audio Signal is Produced which is transformed by Speaker into a audible...

Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...

Hello, I have one conceptual question. I have been working on Supersymmetry. Now, I understand that twice of supersymmetric transformation is equivalent to translation mathematically(naively). However, I don't quite understand why this should be the case conceptually. Supersymmetric...

Hello i have to find the Lorentz transformation for arbitrary velocity (v) relative to (O) the information's i have: 1-i have to use all 3 components of velocity ##(V_x, V_y, V_z )## 2- ##x'=\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}## ##y'=y## ##z'=z## 3-...

Consider an integral of the type ## \int_0^{a} \int_0^{\pi} g(\rho,\varphi,\theta) \rho d\varphi d\rho ##. As you can see, the integral is w.r.t. cylindrical coordinates on a plane but the integrand is also a function of ##\theta## which is a spherical coordinate. So for evaluating it, there are...

When considering a small beam of null-geodesics in spacetime it is possible to define the solid angle spanned by two of the rays at the observer. At page 111 in "Gravitational Lenses" by P.Schneider et. al. they state with reference to Figure (b) that: "The dependence of this distance on the...

Homework Statement Determine a ##2\times 2## matrix ##\mathbb{S}## that can be used to transform a column vector representing a photon polarization state using the linear polarization vectors ##|x\rangle## and ##|y\rangle## as a basis to one using the circular polarization vectors...

$\textbf{Problem}$ Let $\textbf{b} = \begin{bmatrix}\begin{array}{r} 8 \\ 7 \\ 5 \\ -3 \end{array}\end{bmatrix}$ and let $A = \begin{bmatrix} 2 & 3 & 5 & - 5 \\ -7 & 7 & 0 & 0 \\ -3 & 4 & 1 & 3 \\ -9 & 3 & -6 & -4 \end{bmatrix}$ Is $\textbf{b}$ in the range of the transformation $\textbf{x}...

I am solving a nonlinear ODE in the form of Newton's Second Law. In the equation, there is a Heaviside Theta Function of the function which I am solving (##\theta (x(t)##). Since it is quite troublesome to have both the left side of the ODE and the imput of the ODE to contain function of unknown...

This is a solution that I observed from my textbook to a linear transformation problem: Isn't $T$ not linear since $\textbf{x} \ne \textbf{0}$? Property iii of the Definition of Linear Transformation states $T(\textbf{(0)} = \textbf{0}$ so something is contradictory here.

$\textbf{Problem:}$ Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation that reflects each point through the $x_2$ axis. Make two sketches that illustrate properties of linear transformation. $\textbf{Solution:}$ Let $T(\textbf{x}) = \begin{bmatrix} -1 & 0 \\ 0 & 1...

$\textbf{Problem}$ Let $\textbf{u}$ and $\textbf{v}$ be vectors in $\mathbb{R}^n$. It can be shown that the set $P$ of all points in the parallelogram determined by $\textbf{u}$ and $\textbf{v}$ has the form $a\textbf{u} + b\textbf{v}$, for $0 \le a \le 1, 0 \le b \le 1$. Let $T: \mathbb{R}^n...

$\textbf{Problem}$ Let $\textbf{u}$ and $\textbf{v}$ be linearly independent vectors in $\mathbb{R}^3$, and let $P$ be the plane through $\textbf{u}, \textbf{v}$ and $\textbf{0}.$ The parametric equation of $P$ is $\textbf{x} = s\textbf{u} + \textbf{v}$ (with $s$, $t$ in $\mathbb{R}$). Show that...

How do you prove that rotation of a vector is a linear transformation? It's intuitive (although not completely crystal clear to me) that it is a linear transformation at the 2d level, but how do I prove it to myself (that this is a general property of rotations)? For example, rotate vector...

$\textbf{Problem}$ The line segment from $\textbf{p}$ to $\textbf{q}$ is the set of points of the form $(1 - t)\textbf{p} + t\textbf{q}$ for $0 \le t \le 1$ (as shown in the figure below). Show that a linear transformation, $T$, maps this line segment onto a line segment or onto a single point...

$\textbf{Problem}$ Given $\textbf{v} \ne \textbf{0}$ and $\textbf{p}$ in $\mathbb{R}^n$, the line through $\textbf{p}$ in the direction of $\textbf{v}$ is given by $\textbf{x} = \textbf{p} + t\textbf{v}$. Show that linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps this line...

Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear transformation that is violated when $b = 0$ $\textbf{c.}$ Why is $f$ called a linear function?

Homework Statement Homework Equations included in the first picture The Attempt at a Solution i feel confident in my answer to part "a". i pretty much just did what the u and v example at the top of the page did. but for part "b" i tried to distribute and collect like terms and what not...

Homework Statement From Hoffman and Kunze: Is there a linear transformation T from R^3 to R^2 such that T(1,-1,1)=(1,0) and T(1,1,1)=(0,1)?Homework Equations T(c\alpha+\beta)=cT(\alpha)+T(\beta) The Attempt at a Solution I don't really understand how to prove that there is a linear...

How do you transform a curvy coordinate system to that in euclidean space? An example will be greatly appreciated.

V′μ=((∂yμ)/(∂xν))*Vν This is a contravariant vector transformation. (Guys I am really sorry for making the formula above looks so incomprehensible as I still new to this.) For the y in the partial derivative, is y a function in terms of x? In that sense, is it formula that maps x to y? Is it...

To prove the wye-delta transformation formula, it is said 'If the two circuits are to be equivalent, the total resistance between any two terminals must be the same.' But why ? I can't convince myself that it is sufficient condition for the equivalence of circuits.

Homework Statement For the balanced three-phase loads shown in FIGURE 3, ZY = (15 + j15) Ω and ZΔ = (45 + j45) Ω. Determine: Uploaded file C1.png (a) the equivalent single Δ-connected load, (b) the equivalent single Y-connected load obtained from the Δ-Y transformation of (a) above, (c) the...

As known, any Lorentz transformation matrix ##\Lambda## must obey the relation ##\Lambda^μ{}_v####\Lambda^ρ{}_σ##gμρ=gvσ . The same holds also for the inverse metric tensor gvσ which has the same components as the metric tensor itself (don't really understand why every tex formula starts from a...

Homework Statement If two particles have velocities u and v in frame S, find their relative speed in frame S'. Homework EquationsThe Attempt at a Solution Isn't it strange that the relative speed doesn't depend on the velocity of the frame, ##\vec s##? Since the two particles have velocities...

Homework Statement Use the Lorentz Transformation equations to derive the formula relating the time period of a moving clock to that of a stationary clock Homework Equations X'=y(X-vt) Y'=Y Z'=Z t'=y(t-vx/c^2) The Attempt at a Solution t'=1/sqrt(1-(v/c)^2) . (t-vx/c^2)

I understand the concepts behind the terms in the title; however, I have a question about how to transform the wave energy itself. I'm working on a science fair project that involves transforming sound energy into electrical energy--I understand this is not a very reasonable method of harvesting...

Hi all,I am trying to understand relativity and Lorentz Transformation more clearly but I have some problems. Assume that we have frame F' which is moving at velocity v with respect to F. Now assume we have an object, O, moving at velocity, w, with respect to F. Frame F has its own time, t, and...

Homework Statement Homework EquationsThe Attempt at a Solution a) the question is to transform star to delta ?

hi, i know unitary transformation - but could not get where do we need finite and infinite unitary transformation ? please help me in this regard. thanks

I'm trying to derive (14.25) in B&J QFT. This is ##U(\epsilon)A^\mu(x)U^{-1}(\epsilon) = A^\mu(x') - \epsilon^{\mu\nu}A_\nu(x') + \frac{\partial \lambda(x',\epsilon)}{\partial x'_\mu}##, where ##\lambda(x',\epsilon)## is an operator gauge function. This is all being done in the radiation...

r\rightarrow r-2qz and \psi\rightarrow\psi+q\cdot(r-qz), I don't know how to derive it, anybody know? This question results from the book "Optical Solitons: From Fibers to Photonic Crystals [1 ed.]" section 6.5

Sorry for such a simple question but where do we model the energy going during phase transitions? If I had a mercury thermometer in a pot of water, and I had a 200 degree Celsius heat reservoir in contact with the water, I would see the water temp hold steady during the phase transition...

Hi everyone, I am having some problems understanding Bergmann's problems. Problem 3 from Chapter 4 from Intro to the Theory of Relativity by Bergmann 1. Suppose that the frequency at a light ray is f with respect to a frame of reference S. Its frequency f′ in another frame of reference, S'...

If we have: $$F_{\mu\nu} \rightarrow \cos\alpha F_{\mu\nu} +\sin\alpha \star G_{\mu\nu}$$ $$G_{\mu\nu} \rightarrow \cos\alpha G_{\mu\nu} +\sin\alpha \star F_{\mu\nu}$$ for rotation $\alpha$. If infinitesimal transformation for small alpha one gets $$\delta F_{\mu\nu} = \delta\alpha~\star...

I should mention that I'm self-studying this material, not taking it as part of a course, but since this is still a homework-style problem I figured it'd be best to post here. Homework Statement In Peskin and Schroeder problem #11.2, they ask us to consider the Lagrangian: $$\mathcal{L} =...

if A is a square matrix, and A' = B-1AB is its similarity transform (with a non-singular similarity transformation matrix B), then the eigenvalues of A and A' are supposed to be the same. I can verify this for all most all cases of A. But, it doesn't seem to work, when the eigen values of A are...

I was reading in this book: Supergravity for Daniel Freedman and was checking the part that has to do with Extremal Reissner Nordstrom Black Hole. He was using killing spinors (that I am very new to). I was understanding the theory until he stated with the calculations: He said that the...