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
for L, M: V -> W, L, M, linear let||L|| = sup{|L(v)|: v in V, |v| <= 1}
show ||L + M|| < ||L|| + ||M||
Homework Equations
The Attempt at a Solution
so is it true that if |L(x) + M(x)| defines a sup for L + M (x for which |L(x) + M(x)| is the sup), then it also defines a...
Homework Statement
Let V be the vector space of all functions f: R->R which can be differentiated arbitrarily many times.
a)Let T:V->V be the linear transformation defined by T(f) = f'. Find the (real) eigenvalues and eigenvectors of T. More precisely, for each real eigenvalue describe the...
What is the common name for norm-preserving linear transformations in a normed linear space? I want to say they are the unitary transformations, but I'm just fuzzy enough not to know a good way of proving it.
i need some help with this question -
lets say if A =
|val1 val2 |
|val3 1 |
what would AA^t equal?
and
AA^t and A^T.T are symmetrical. is this true for any 2x2 matrix?
thanks in advance
Homework Statement
A linear transformation L : R2 -> R3 is defined by:
L({\bf{x}}) = \left( {x_2 ,x_1 + x_2 ,x_1 - x_2 } \right)^T
I wish to find the matrix representation of L with respect to the orderes bases [u1, u2] and [b1, b2, b3], where
u1 = (1,2)
u2 = (3,1)
andb1 = (1,0,0)
b2 =...
Homework Statement
It is often stated that many forms of transport transform chemical energy into kinetic energy. Explain why a cyclist traveling at constant speed is not making this transformation. Explain what transformations of energy are taking place.
The Attempt at a Solution
1...
Homework Statement
T:{R^3 \rightarrow {R^2} given by T(v_1,v_2,v_3) = (v_3 -v_1, v_3 - v_2)
If linear, specify the range of T and kernel T
The attempt at a solution
Okay, I went ahead and tried to find the kernel of T like here:
\begin{align*}&v_3 - v_1 = 0\\
&v_3 - v_2 =...
there is this table that i showed in my previos question
now i need to implement it using 16X4 and 4 X4 proms
here is the table:
http://s290.photobucket.com/albums/ll279/transgalactic/?action=view¤t=IMG_8814.jpg
here is the table...
[SOLVED] Linear transformations
Homework Statement
Determine whether the following maps are linear transformations. (proofs or counterexamples required)
a.) L: R^2\rightarrowR^2,
(x1)
(x2)
\mapsto
(2x1 + 3x2)
(0)
The brackets should be two large brackets surrounding the two...
Sorry to bring up again a question that I asked before but I am still confused about this.
In SR we have Lorentz invariance.
Now we go to GR and one says that the theory is invariant under general coordinate transformations (GCTs). But, as far as I understand, this is simply stating that...
I'm trying to implement AES as practice for my C++ skills, but I've come across a confusing problem that I think belongs here rather than in programming.
Rijndael's finite field is GF(28), with reducing polynomial x8+x4+x3+x+1
There is a step in the algorithm that takes a polynomial...
Hi, New here...Can't seem to do latex on here so this post is incomplete until I can work it out.
This is maybe quite abstract and generic, but here goes. This problem has niggled me for a while and I need some input please.
I have an action S=\int d^4 x \sqrt(g(x))\overline\Phi...
Hello all.
I asked this question as a sub-question in another thread where it was perhaps inappropriate. It is very basic but the more i try to understand relativity the nearer to the absolute basics i need to go. The more i learn the less i seem to actually understand.
When length and...
Homework Statement
Masses of 350g and 175g are attached by a light string and hanging straight down from a light frictionless pulley. The 350g mass is 1.5m above the ground. What speed will the system have when the 350g mass hits the ground.
My attempt at a data list is (after i drew a...
I find in the literature the following transformation equations for the space-time coordinates
x'=g(x-vt)
t'=t/g
g=gamma.
Please tell me what do they bring new in the approach to SRT?
Thanks
[SOLVED] Linear algebra - transformations
Homework Statement
Please take a look at:
http://www.math.luc.edu/~jdg/w3teaching/math_212/sp02/PDF/test2practice.pdf
Please take a look at #7, question c. To determine if the vector w is in the image (range) of T, I find the matrix B that represents...
[SOLVED] Combined linear transformations
Homework Statement
I have a linear transformation L : R^3 -> R^3 represented by a matrix A. I also have another linear transformation S : R^3 -> R represented by a matrix B.
The dimensions of the matrix A must be 3x3 and for B it is 1x3. I have to find...
[SOLVED] Linear algebra - transformations
Homework Statement
I actually have two questions:
1) I have a linear transformation L and it is represented by a matrix A. I also have a vector w, and I want to find out if w gets "hit" by L - see "answer-part" for my approach, and please comment.
2)...
Homework Statement
I have a transformation (not linear! that is what I have to show) F given by:
F : P_4 -> P_7 (P_7 is the vector-space spanned by polynomials less than degree 7). I also know that F(p(x)) = (p(x))^2.
The matrix A representing F with respect to the two basis is the one I...
Can anyone explain how to derivate "Supersymmetric Transformations" like \phi\rightarrow\psi??
It seems to me that there's no symmetry at all between bosons and fermions.
Can anybody know any proofs??
I have been working in the properties of the large gauge transformation of QCD in the temporal gauge and I have shown that these satisfy U_{n}U_{m} and commutes with the translations where the large gauge transformations U_n and U_m belongs to the homotopy classes characterized by winding...
A quick question this time...
Example: Let (u,v)=f(x,y)=(x-2y, 2x-y).
Find the region in the xy-plane that is mapped to the triangle with vertices (0,0),(-1,2),(2,1) in the uv-plane.
Solution:
(0,0)=f(0,0), (-1,2) = f(5/3,4/3), and (2,1)=f(0,-1), the region is the triangle with...
A boy stands at the peak of a hill which slopes downward uniformly at angle \phi . At what angle \theta from the horizontal should he throw a rock so that is has the greatest range.
Ok, so this is a rotation of the normal x_{1} - x_{2} plane right? So we can use the direction cosines...
Hello,
Can someone help me with this problem? Thanks in advance
Let T be a linear transformation such that T (v) = kv for v in R^n.
Find the standard matrix for T.
1) http://www.geocities.com/asdfasdf23135/advcal13.JPG
Let F1 = x^2 - y^2 + z^2 -1 = 0
F2 = xy + xz - 2 = 0
F3 = xyz - x^2 - 6y + 6 = 0
My thought is to compute the gradients, grad F1 and grad F2. Then by taking their cross product, I can get a tangent vector v for the curve. Now, I can feel...
[SOLVED] Linear Transformations (polynomials/matrices)
Never mind, I can see it now, thanks
Homework Statement
Let S be the linear transformation on P2 into P3 over R. S(p(x)) = xp(x)
Let T be the linear transformation on P3 over R into R2x2 defined by T(a0 + a1x + a2x^2 + a3x^3) = [ a0 a1...
Homework Statement
let T: R^{3} -> R^{3} be the mapping that projects each vector x = (x(subscript 1) , x(subscript 2) , x(subscript 3) ) onto the plane x(subscript 2) = 0. Show that T is a linear transformation.
Homework Equations
if c is a scalar...
T(cu) = cT(u)
T(u + v) = T(u) +...
Homework Statement
Find a basis for the image of the linear transformation T: R^4 -->R^3 given by the formula T(a,b,c,d) = (4a+b -2c - 3d, 2a + b + c - 4d, 6a - 9c + 9d)
Homework Equations
The Attempt at a Solution
Well this question followed asking about the basis for the kernel...
Homework Statement
I have a questions on Neutrino Oscillations, but i have no idea how to solve any of it... this is just one part...
How do I show that the probability of a mu neutrino having transformed into a tau neutrino at a time t is:
sin^2(2theta)sin^2[((difference in masses...
Homework Statement
Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace. T is the transformation on R^3 that rotates points about some line through the origin.
Homework Equations
maybe...Ax=(lambda)x ?
The Attempt...
Describe the transformation on the graph of y=1/x needed to obtain the graph of each of the following:
a) y= x+3/x+1
b) y= 2x/x-1
im stuck on how to answer this question...how would i solve this?...thanks
Homework Statement
Find the inverse Laplace transform of the given functions:
3. \frac{2}{s^2+3s-4}
7. \frac{2s+1}{s^2-2s+2}Homework Equations
Inverse Laplace Transform TableThe Attempt at a Solution
on 3. i made the denominator look like (s+4)(s-1) but i got lost from there. i couldn't find...
Homework Statement
I am trying to show that
\vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b
where the e's are bases on a manifold and the primes mean a change of coordinates
I can get that \frac{\partial x^a}{ \partial x'^b} dx'^b \vec{e}_a = dx'^a \vec{e'}_a from the invariance...
1. Graph the function f(x)=x^2+4x+3 by starting with the graph of y=x^2 and using transformations.
2.
3. I know the graph opens up, but I don't understand transformations or how to solve them, any help would be greatly appreciated.
Hi. I thought I had tensors and Lorentz transformations under control, but now I'm in doubt again.
For example, consider the electromagnetic field tensor
F_{\mu\nu} = \begin{pmatrix}
0 & -E_1 & -E_2 & -E_3 \\
E_1 & 0 & B_3 & -B_2 \\
E_2 & -B_3 & 0 & B_1 \\
E_3 & B_2...
Homework Statement
A physics professor on Earth gives an exam to her students who are on a spaceship traveling at speed v relative to Earth. The moment the ship passes the professor she signals the start of the exam. If she wishes her students to have time To (spaceship time) to complete the...
Please have a critical look at the lines below:
The simplest derivation of the Lorentz transformation simplified: J.M.Levy "A simple derivation of the Lorentz transformation and of the accompanying velocity and acceleration changes," Am.J.Phys 35,615 (2007) arXiv:physics/0603103 revisited.[1]...
divergence question
show that the divergence transforms as a vector under 2D rotations.
I am so confused abouth what this question wants me to do. Obviously the divergence is not invariant under rotations. Consider the divergence of the function f(x,y) = x^2 * x-hat. The divergence is...
Homework Statement
I am trying to learn from Srednicki's QFT book. I am in chapter 2 stuck in problem 2 and 3. This is mainly because I don't know what the unitary operator does - what the details are.
Starting from:
U(\Lambda)^{-1}U(\Lambda')U(\Lambda)=U(\Lambda^{-1}\Lambda'\Lambda)
How does...
lets say you apply a Lorentz boost in the x direction with velocity v and a Lorentz boost in the y direction with velocity v'. Why does it makes that the order in which you apply the transformations affects the resultant transformation matrix? These are two independent directions, so shouldn't...
I've been discussing some things with Samalkhaiat over in the conformal
field theory tutorial. A part of that conversation (indicated by the new
title) was drifting away from CFT matters, so we both thought it was better
to move it into the Quantum Physics forum, to minimize pollution of the...
Can someone explain to me what it means to be "covariant" in the context of special relativity and Lorentz transformations? I already checked wikipedia.
Hi all.
I have seen a lot of different forms of the KdV equation...
The derivation of it results in a form like
Ut+Ux+epsilon(UUx+Uxxx)=0
and after some transformation, the epsilon is removed the equation becomes
Ut+Ux+UUx+Uxxx=0, and, still, after some sort of transformation, it becomes...
Homework Statement
I have a general question. If we have some subspace W of R^n where dimW=k. Then if T is an orthogonal transformation from R^n->R^n is the dimension of T(W) also k?
Homework Equations
The Attempt at a Solution
The reason I think this is true is because if...
Find an orthogonal transformation T from R3 to R3 such that
T of the column vector [2/3 2/3 1/3] is equal to the column vector [0 0 1]
So I tried to construct out the 3x3 matrix
[a b c]
[d e f]
[g h i]
and applied the properties of an orthogonal matrix and basic algebra. I ended up with a...
If x,y of R^n (as a normed vector space) are non-zero, the angle between x and y, denoted <(x,y), is defined as arccos x.y/(|x||y|).
The linear transformation T :R^n----->R^n
is angle preserving if T is 1-1, and for x,y of R^n (x,y are non zero) we have
<(Tx,Ty) = <(x,y).
what are...
If x,y of R^n (as a normed vector space) are non-zero, the angle between x and y, denoted
<(x,y), is defined as arccos x.y/(|x||y|).
The linear transformation T :R^n----->R^n
is angle preserving if T is 1-1, and for x,y of R^n (x,y are non...
Homework Statement
Given two sub-spaces of R^n - W_1 and W_2 where dimW_1 = dimW_2 =/= 0.
Prove that there exists an orthogonal transformation T:R^n -> R^n so that
T(W_1) = T(W_2)
Homework Equations
The Attempt at a Solution
If dimW_1 = dimW_2 = m then we can say that...