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
Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix}
a & 1 & 0 \\
b & 0 & 1 \\
c & 0...
I don't understand why we can write the elements of the lorentransformation in the form
## {\Lambda}^{\mu}\:_{\nu} = [exp(-\frac{i}{2}{\omega}^{\rho\sigma}M_{\rho\sigma})]^{\mu}\:_{\nu} ##
I know that we can write it in the form
## {\Lambda} = exp(t\Theta) ##
where
## \Theta ##
are elements...
Homework Statement
[/B]
In Minkowski spacetime we are considering a (series of) frame(s), S', attached to a rocket with constant proper acceleration. The rocket's speed in S is v.
We find with boundary conditions x = 0 at t = t' = 0 the relationships between S and S' (for x' = 0, i.e. at the...
Hey,
I need help with part D2. My explanation is not right so I honestly do not know what I am suppose to write. My assignment is attached to this thread.
Homework Statement
T: R^2 --> R^2 given as a projection on the line L = 5x+2y=0
decide matris T?
Homework EquationsThe Attempt at a Solution
L= 5,2
X=x1, x2
projL on X = (5x1+2x2)/29 *(5,2)
= 1/29 [25 10
10 4]
is this correct?
Homework Statement
Derive the Lorentz Transformation using light cone coordinates defined by
##x^±=t±x##
##x^+ x^-~## is left invariant if we multiply ##~e^φ~## to ##~x^+~## and ##~e^{-φ}~## to ##~x^-~##, that is ##~x'^+ x'^-=x^+ x^-##
Homework Equations
##t'^2 - x'^2 = t^2 - x^2...
the question is: Rewrite the rational equation y=(-5x-18)/(x+4) to show how it is a transformation of y=1/x. describe transformations
looks like it is shifted 4 to left, then stretched by factor of -5x-18. Is that accurate? would you elaborate beyond that?
1. Show that the map $\mathcal{A}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ defined by $\mathcal{A}(x,y,z) = (x+y, x-y, z)$ is a linear transformation. Find its matrix in standard basis.
2. Find the dimensions of $\text{Im}(\mathcal{A})$ and $\text{Ker}(\mathcal{A})$, and find their basis for the...
I'm confused about the notation
T:R^n \implies R^m
specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?
Homework Statement
Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question.
Consider the...
Homework Statement
Consider the linear transformation T from
V = P2
to
W = P2
given by
T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2
Let E = (e1, e2, e3) be the ordered basis in P2 given by
e1(t) = 1, e2(t) = t, e3(t) = t^2
Find the coordinate matrix...
Homework Statement
Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2.
Homework Equations
-Eigenvalues and...
Hi. In the attached proof for Lorentz transformation for momentum http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/lorentz_transformation_E_p.pdf, there is this step that I don't understand:
1/√1-u'2/c2 = γ(1-vux/c2)/√1-u2/c2
Can someone explain how they derived this? Thanks! :)
Taking a look at "http://www.space.com/30026-earth-twin-kepler-452b-exoplanet-discovery.html" I observe that planet Kepler-452b (judged to be somewhat Earth-like) is 1400 light years from Earth. If a spaceship leaves Earth at a fifth of the speed of light, traveling toward Kepler-452b, from...
Dear all,
I'm taking a second look at the Newtonian limit of GR and the covariance group. My main interest is to see how the Newton potential transforms under the covariance group of Newtonian gravity. I know that the gradient of the potential is given by the Christoffel symbol \Gamma^i_{00}...
Homework Statement
dy/dx = (2x +y -1) / ( 4x -2y +1) , x= X +1 , y = Y-1 ,, how to make it into differential equation ? my ans is not same as the ans given .
P/s : in the second photo , it's lnx +c , sorry for the blur photo
Homework EquationsThe Attempt at a Solution
I'm reading A. Zee's GR book and I'm in the section in which he is showing how to transform coordinates to be locally flat in a neighborhood of a point.
He said that we can always choose our neighborhood to be locally flat for any space of any dimension D.
"Look at how the metric transforms...
Homework Statement
The given problem is that we have a rocket ship, accelerating at a constant rate of 1g (in its own instantaneous inertial rest frame) for 40 years. We must find the distance it travels in that time, as measured by an observer on earth.
Homework Equations
dx'=gamma*(dx-vdt)...
Homework Statement
Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...
I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems.
The contravariant transformation formula is:
Vj = (∂yj/∂xi) * Vi
where Vj is in the y- frame of reference and Vi is in the x-frame of reference. Einstein summation...
Homework Statement
Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##.
Homework Equations
##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy##
##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...
Homework Statement
The image of a linear transformation = columnspace of the matrix associated to the linear transformation.
More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X...
Hi..
I am new to this forum and not sure whether this is the right place to ask for a help.
I have to transform angular velocity vector of a satellite from Earth Centered Inertial (ECI) coordinate system to Local Vertical Local Horizontal(LVLH) system. How can I do that..? Any help appreciated..
When a test charge is stationary, it generates an E field with zero curl.
When it moves, it generates a circulating magnetic field B...( as well as the E field.)
Is there a geometric reason for this? How does motion alone generate a circulating field B?
Can we call this change a...
I have a left-handed ##SU(2)## lepton doublet:
##
\ell_L = \begin{pmatrix} \psi_{\nu,L} \\ \psi_{e,L} \end{pmatrix}.
##
I want to know its transformation properties under conjugation and similar 'basic' transformations: ##\ell^{\dagger}_L, \bar{\ell}_L, \ell^c_L, \bar{\ell}^c_L## and the general...
Homework Statement
We now specify the velocity v to be along the positive x1-direction in S and of magnitude v. We also consider a frame \overline{S} which moves at speed u with respect to S in the positive x1-direction.
question 1 : Write down the transformation law for p^\mu .
question 2...
Homework Statement
Being ##f : \mathbb R^4\rightarrow\mathbb R^4## the endomorphism defined by:
$$f((x,y,z,t)) = (13x + y - 2z + 3t, 10y, 9z + 6t, 6z + 4t)$$
1) Determine the basis and dimension of ##Ker(f)## and ##Im(f)##. Complete the base chosen in ##Ker(f)## into a base of ##\mathbb R^4##...
Homework Statement
1.) Prove that the infinitesimal volume element d3x is a scalar
2.) Let Aijk be a totally antisymmetric tensor. Prove that it transforms as a scalar.
Homework EquationsThe Attempt at a Solution [/B]
1.) Rkh = ∂x'h/∂xk
By coordinate transformation, x'h = Rkh xk
dx'h =...
The thought experiment used to prove Lorentz transform uses a light signal as an assumption. What if there was something other than the light signal then Lorentz transformation would have totally different term in place of 'c'(speed of light).
Homework Statement
Question: What is the defect of a linear transformation?
2. The attempt at a solution
A defective matrix (of a linear transformation) is a matrix that doesn't have a complete basis of eigenvectors.
Does this mean that linearly dependent vectors of a matrix are called defects?
Homework Statement
What can be special cases, in the study of electrical circuits, in which one can not apply wye delta transformation? Can someone graph them in some way?
Homework EquationsThe Attempt at a Solution
Iam having trouble understanding how one arrives at the transformation matrix for spherical to rectangular coordinates.
I understand till getting the (x,y,z) from (r,th
ie.,
z = rcos@
y = rsin@sin#
x = rsin@cos#
Note:
@ - theta (vertical angle)
# - phi (horizontal angle)
Please show me how...
Homework Statement
Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y =...
Evening all,
I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y = -m x -b are mapped...
I have seen this calculation on web
Let consider that old man displaced the cart from pole A to Pole B on platform. Observers are on platform S & in train S', moving with velocity –V then (let, X-axis is parallel to train direction)
1) When AB (displacement) parallel to the direction of train...
Homework Statement
Let A(l) =
[ 1 1 1 ]
[ 1 -1 2]
be the matrix associated to a linear transformation l:R3 to R2 with respect to the standard basis of R3 and R2. Find the matrix associated to the given transformation with respect to hte bases B,C, where
B = {(1,0,0) (0,1,0) , (0,1,1) }
C =...
Homework Statement
Prove that there exists only one linear transformation l: R3 to R2 such that:
l(1,1,0) = (2,1)
l(0,1,2) = (1,1)
l(2,0,0) = (-1,-3)
Find Ker(l), it's basis and dimension. Calculate l(1,2,-2)
Homework EquationsThe Attempt at a Solution
I still find linear transformations...
Homework Statement
Can anybody suggest hints on how to show that x'=xcosΘ-ysinΘ, y'=xsinΘ+ycosΘ by using the infinite string of poisson brackets?
Homework Equations
ω→ω+a{ω,p}+a^2/2!{{ω,p},p}+...
The Attempt at a Solution
Sorry, I just can’t think of any way, substituting doesn’t work.
In obtaining (5.362) from (5.359), we first get
##U_{\tau}i\hbar\frac{\partial}{\partial t}\Psi(t) = U_{\tau}H^*\Psi^*(-t)##
In order to obtain the LHS of (5.362), ##U_{\tau}## must commute with ##\frac{\partial}{\partial t}##. But how do we know that they commute?
Spinors in $N=2, D=4$ supergravity can be simplified using gauge transformation and thus canonical spinors can be found. In the case of $N=2, D=4$ supergravity the gauge transformation Spin (3,1) is used. My question is how do we know which transformation can be used in a certain theory in order...
Homework Statement
Is the transformation ##Y:(t,x,y,z)\rightarrow (t,x,-y,z)## a Lorentz transformation? If so, why is it not considered with P and T as a discrete Lorentz transformation? If not, why not?
Homework Equations
The Attempt at a Solution
A Lorentz transformation ##\Lambda##...
How does it work? (The derivative rules of FT)
We look at $$F[x(t)]=\hat{x}(f)$$
$$\mathcal{l} \text{ is a distribution, with}\tilde{x}=tx(t)$$
$$\mathcal{F}[Dl(x)]=\mathcal{F}l'(x)=2\pi il(\mathcal{F}\tilde{x})=2\pi i \mathcal{F}l(\tilde{x})$$
Till here I fully understand. But next step...
In Newmann-Penrose formalism, a Null rotation with ##l## fixed is
$$l^a−>l^a\\
n^a−>n^a+\bar{c}m^a+c\bar{m}^a+c\bar{c}l^a\\
m^a−>m^a+cl^a\\
\bar{m}^a−>\bar{m}^a+\bar{c}l^a$$
Using this transformation, how to prove?
$$π−>π+2\bar{c}ϵ+\bar{c}^2κ+D\bar{c}$$
Ref: 2-Spinors by P.O'Donell, p.no, 65
Hi All;
I was trying to understand Lorentz Transformation equation and special theory of relativity, but as I compared the derivation with a thought experiment which I imagined I found the whole Lorentz Transformation Equation fails. The details of the problem is given below. I know I m wrong...
Hi All;
I was trying to understand Lorentz Transformation equation and special theory of relativity, but as I compared the derivation with a thought experiment I created I found the whole Lorentz Transformation Equation fails. The details of the problem is given in the pdf file attached. I know...
Homework Statement
Given the transformation fh : R 3 → R 3 defined by fh(x, y, z) = (x−hz, x+y −hz, −hx+z), where h ∈ R is a parameter.
a) Find, for all possible values of h, Ker(fh), Im(fh), their bases and dimensions.
b) Is fh an isomorphism for some value of h?
Homework Equations
Ax=o
The...
Alternate title: Is the textbook contradicting itself?
imgur link: http://i.imgur.com/3sTVgwr.jpg
But
imgur link: http://i.imgur.com/33Ufncb.jpg
So...it would appear that transposing has the property of linearity, but no matrix can achieve it...is transposing a linear transformation? The...
Mod note: Moved from Precalc section
1. Homework Statement
Given l : IR3 → IR3 , l(x1, x2, x3) = (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3, x1 + x2 + x3), find Ker(l), Im(l), their bases and dimensions.
My language in explaining my steps is a little sloppy, but I'm trying to understand the process and...