What is Transformations: Definition and 861 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.

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1. I The Hamiltonian and Galilean transformations

In a classical example, for a system consisting of a mass attached to a spring mounted on a massless carriage which moves with uniform velocity U, as in the image below, the Hamiltonian, using coordinate q, has two terms with U in it. But if we use coordinate Q, ##Q=q−Ut##, which moves with the...
2. A Are equations of motion invariant under gauge transformations?

We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations? If yes, can you show a mathematical proof (instead of just saying in words)?
3. Transformations to both sides of a matrix equation

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8. Canonical transformations of a quantized Hamiltonian?

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9. I Coordinate and time transformations

In describing the Galelian or Lorentz transformations, All books I've read keep talking about clocks and meter sticks, but I don't see how an event happening away from the observer could be instantaneously described by a set of coordinates and a point in time; information conveying the event...
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11. I The different generators of canonical transformations

Consider the phase space of a one degree of freedom mechanical system. We can pass from one phase space coordinates to another phase space coordinates via a canonical transformation. I want to focus on 1-parameter canonical transformations, $$(q_{0},p_{0})\rightarrow(q_{\lambda},p_{\lambda})$$...
12. B Galilean vs Lorentz Transformations: Correct Understanding?

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18. Energy transformations in an IC engine cylinder

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19. MHB Matrix Transforms: nxm, n->m, m->n, n+m->n/m

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20. I Angle-Preserving Linear Transformations in 2D Space for Relativity

I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the ##ct## axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the ##y## axis. When we switch...
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22. I Another interpretation of Lorentz transformations

I consider three material points O, O', M, in uniform rectilinear motion in a common direction, so that in relation to the point O, the points O' and M move in the same direction with the constant velocities v and u (u>v>0). Assuming that at the initial moment (t0=0), the points O, O', M were in...
23. Proper Lorentz transformations from group theory?

Hi, I was looking at this derivation https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates and I was wondering 1- where does the group structure come from? The principle of relativity? or viceversa? or what? 2- why only linear transformations? I remember...
24. I Trouble understanding contravariant transformations for vectors

Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system. For instance, let's say we have two coordinate...
25. T

I Time and Lorentz transformations

Hello, why time is the fourth dimention and not another quantity or variable? General relativity has as a special case the special relativity, so Lorentz transformations are contained in general relativity but are they in a more general form than that of special relativity generally? If they...
26. MHB 5.2a plot linear transformations

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27. I Lorentz transformations: 1+1 spacetime only

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28. B Understanding Lorentz Transformations in Special Relativity

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29. MHB Row and column transformations

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31. Lorentz transformations for electric and magnetic fields

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32. E

Composing a few transformations

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33. E

B Unitary transformations

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34. Graphical Transformations and Finding the Equation of a Curve

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42. B What are matrix transformations?

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44. I Given two linear transformations L and K, show ##K = \lambda L## holds

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45. Thevenin's Theorem -- Using transformations to find the equivalent resistance

Hello Guys, I really need help, I am trying to simplify this circuit to calculate Rth and I got stuck.
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48. MHB Complex-Linear Matrices and C-Linear Transformations .... Tapp, Propostion 2.5 .... ....

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49. I Complex-Linear Matrices & C-Linear Transformations .... Tapp, Propn 2.4 ....

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focused on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices". I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4. Proposition 2.4...
50. I Lewis H Ryder: Cartesian to Polar Coord Transformations

The example is about the transformation between the cartesian coordinates and polar coordinates using the definition In lewis Ryder's solution, I got confused in this specific line I really can't see how is that straightforward to find?