What is Transformation: Definition and 1000 Discussions
In linear algebra, linear transformations can be represented by matrices. If
T
{\displaystyle T}
is a linear transformation mapping
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
and
x
{\displaystyle \mathbf {x} }
is a column vector with
n
{\displaystyle n}
entries, then
T
(
x
)
=
A
x
{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some
m
×
n
{\displaystyle m\times n}
matrix
A
{\displaystyle A}
, called the transformation matrix of
T
{\displaystyle T}
. Note that
A
{\displaystyle A}
has
m
{\displaystyle m}
rows and
n
{\displaystyle n}
columns, whereas the transformation
T
{\displaystyle T}
is from
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.
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...
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...
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...
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...
In SR the speed transformation formula (in response to a change of inertial frame of reference) is usually derived from the Lorentz transformation of space and time coordinates. I would like to find a direct derivation starting from the existence of a maximum speed limit (c) in respect to any...
Homework Statement
t:P_3 -----> P_3
p(x) |---> p(x) + p(2)
Determine whether or not this function is linear transformation or not.
Homework Equations
For a function to be a linear transformation then t(0) = 0 , there are other axioms that must be satisfied, but that is not the problem...
My local PBS station broadcasts a physics series on a sub-channel. They call it The Mechanical Universe locally.
PBS recently broadcast The Lorentz Transformation . It appears that one may view these on line, as the Lorentz Transformation began to load after I allowed it to pop-up. There are...
Hey guys, i did this source transformation as an alternate method to find the transfer function of a circuit, however I am getting a different transfer function of 2/(2s+(s+3)(s^2+1)) to the solution in the following image. Any help would be really appreciated :)
Homework Statement
[/B]
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...
We have a r.v. X with p.d.f. = sqrt(θ/πx)*exp(-xθ) , x>0 and θ a positive parameter.
We are required to show that 2 θX has a x^2 distribution with 1 d.f. and deduce that, if x_1,……,x_n are independent r.v. with this p.d.f., then 2θ∑_(i=1)^n▒x_ι has a chi-squared distribution with n...
Homework Statement
let A be the matrix corresponding to the linear transformation from R^3 to R^3 that is rotation of 90 degrees about the x-axis
Homework Equations
find the matrix A
The Attempt at a Solution
I got stuck on rotating z component.
I tried T([e1,e2,e3])=[0 -1 0]...
Was reading how do vectors transform under chiral transformation and found the following:
If $$V^\mu$$ is a vector; set $$ V^\mu = \bar{\psi} \gamma^\mu \psi= $$
$$\bar{\psi}\gamma^\mu e^{-i\alpha\gamma^5}e^{i\alpha\gamma^5}\psi =$$
$$\bar{\psi}\gamma^\mu\psi = V^\mu $$
My questions are why...
Transform the left hand member into the right hand member.
$\frac{\tan\alpha+\tan\beta}{\sec\alpha-\sec\beta}=\frac{\sec\alpha+\sec\beta}{\tan\alpha-\tan\beta}$By using cross multiplication I was able to prove this identity but what I actually want to accomplush is to transform the left member...