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.
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...
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...
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
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...
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)...
Let A be a nxn matrix corresponding to a linear transformation.
Is it true that A is invertible iff A is onto? (ie, the image of A is the entire codomain of the transformation)
In other words, is it sufficient to show that A is onto so as to show that A is invertible?
That was what my...
In McCauley's book Classical Mchanics: Transformations, Flows, Integrable and Chaotic Dynamics we are analyzing a coordinate transformation in order to arrive at symmetry laws. A coordinate transformation is given by q_i(\alpha) = F_i(q_1,...,q_f, \alpha). Then, to the first order Mccauley...
Homework Statement
Consider the transformation from the variables (q,p) to (Q,P) by virtue of q = q(Q,P), p = p(Q,P) and H(q,p,t) = H(Q,P,t). Show that the equations of motion for Q,P are:
\partialH/\partialQ = -JDdP/dt
\partialH/\partialP = JDdQ/dt
where JD is the Jacobian determinant...
Homework Statement
Let A \in M_n(F) and v \in F^n.
Let v, Av, A^2v, ... , A^{k-1}v be a basis, B, of V.
Let T:V \rightarrow V be induced by multiplication by A:T(w) = Aw for w in V. Find [T]_B, the matrix of T with respect to B.
Thanks in advance
Homework Equations...
Homework Statement
In the figure, let S be an inertial frame and let S'
be another frame that is
boosted with speed v along its x'-axis w.r.t. S, as shown. The frames are pictured
at time t = t0 = 0:
A) Find the Non-relativistic transformation (Galilean Transformation) between the two...
Homework Statement
Let va be a dual vector field. Show that the quantity ∂[a vb] transforms as a type (0, 2) tensor under coordinate transformations.
Homework Equations
wu' = (dxu / dxu') wu
The Attempt at a Solution
My main problem is that I don't know what the brackets mean...
Homework Statement
Let F be the cumulative distribution function of a random variable X. Find the cumulative distribution function of Y= {\alpha}X+\beta, where \, \alpha \gt 0
Homework Equations
The Attempt at a Solution
I think this a fairly easy question, I just want to make...
Homework Statement
Could anyone help me please?
I would like to know the proof of the following Laplace transform pair:
Homework Equations
\mathcal{L}_{t \rightarrow s} \left\{ J_0 \left( a\sqrt{t^2-b^2} \right) \right\}=\frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}}
The Attempt at a Solution...