Transformations Definition and 863 Threads

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...

The book I use for relativity states that 4 conditions are required to get the four constants in the lorentz transformations. The 4 conditions the book uses are: 1) velocity of S' seen by S is v. 2) velocity of S as seen by S' is v. 3) Time dilation is same in either frame. 4) Speed of...

Homework Statement Given transformations T_1, T_2:V->F where V is a vector space with the dimension n over the field F, T_1 , T_2 =/= 0. If N_1 = KerT_1 , N_2 = KerT_2 and N_1 =/= N_2 find dim(N_1 intersection N_2) Homework Equations dim(A+B) = dimA + dimB - dim(A intersection B)...

I want to first explain my current understanding and motivation so you guys can whip me into shape in case I'm misunderstanding the starting point -- SR and linear transformations. So, we can write the laws of electrodynamics in terms of the electromagnetic field tensor F^{\alpha \beta} as...

Homework Statement T is a transformation from the vector space of real 2x2 matrices back to that space. T(X) = X - trans(X) (trans = transposed) a)Find a base for KerT and ImT b)Prove that T can be diagonalized. Homework Equations The Attempt at a Solution a) If X is in KerT...

Hey guys This isn't related to a particular question but i thought might be too specific for the general forum so here we go... If you have a function f(x,y) such that u = f(x) and v= g(x) and you have some transformation T(u,v) i know you can find the inverse by getting x and y in terms...

I have been told that using a metric g_{00} = -a^2(\eta)(1+2\psi) g_{oi} = g_{i0} = a^2(\eta)\omega_i g_{ij} = a^2(\eta) \left[(1+2\phi)\gamma_{ij} + 2\chi_{ij} \right] and a gauge transformation x^{\bar{\mu}} = x^{\mu} + \xi^{\mu} with \xi^0 = \alpha \xi^i = \beta^j gives...

Linear transformations... Homework Statement Can't figure these things out for my life. Seriously. Here's an example. consider the basis S={v1, v2, v3} for R^3 where v1=(1,2,1), v2=(2,9,0) and v3=(3,3,4) and let T:R^3-->R^2 be the linear transformation such that: T(v1)=(1,0)...

Question Show that, with V = 4/5c, the Lorentz transformation of the equations, t^prime = y(V) (t-(v/c^2)x) and x^prime = y(V) (x-Vt). (where y(V) = the Lorentz factor). can be written as ct^prime = 5/3ct - 4/3x and x^prime = 5/3x - 4/3ct Relevant equations y(V) = 1/(sqrt1-(V/c)^2) The...

Homework Statement Find a Mobius transformation that maps the real axis to the circle |z-1|=1, and the line Im(z)=1 to the circle |z-2|=1Homework Equations A mobius transformation is one of the form z\rightarrow\frac{az+b}{cz+d} on the extended complex plane The Attempt at a Solution My...

Homework Statement Let T:V  W be a linear transformation. Prove the following results. (a) N(T) = N(-T) (b) N(T^k) = N((-T)^k) (c) If V = W and t is an eigenvalue of T, then for any positive integer k N((T-tI)^k) = N((tI-T)^k) where I is the identity transformation The Attempt at a...

I'm not sure I understand the use of generating functions in canonical transformations. In particular, why are there four basic canonical transformations? It isn't true that any canonical transformation is one of the four basic types, so what makes them special over any other transformation...

Find all matrix transformations f:R^2 -----> R^2 which leave the length of vectors in the plane unchanged Thats R as in the set of all real numbers R. The only possible transformations i could possibly think of that would not change the length is rotation, other than that i am...

Homework Statement Using du=.01, dv=.01 find the aroximate area under the transformation of the square bounded by the lines u=3, u=3.01, v=5, v=5.01. Homework Equations T(u,v)=<au+bv, cu+dv> where a, b, c, and d make a square matrix. The Attempt at a Solution I am not sure...

Incase anyone doesn't understand the notation, GL(2;C) is the group of linear transformations on C^2 which are invertible. Another way of looking at it is all complex 2x2 matrices with non-zero determinant. It is fairly easy to show that GL(2;C) is not simply connected (just define a...

How would you graph y=f(x)-4? I am not sure how the original graph looks like y=f(x) either. Also, if i were to graph this using a graphing calculator, how would that be done?

Homework Statement a) Show that you can split any transformation into a Translation, Dilation-Rotation, Inversion, and Translation. b) Show using part a) that any straight line or circle is send to a straight line or circle when applying the mobius transformation. Homework Equations...

Homework Statement Show that the eigenvectors of a unitary transformation belonging to distinct eigenvalues are orthogonal. Homework Equations I know that U+=U^-1 (U dagger = U inverse) The Attempt at a Solution I tried using a similar method to the proof which shows that the...

How do you derive hypergeometirc identities of the form 2F1(a,b,c,z)= gamma function. What I mean is that the hypergeometric function converges to a set of gamme functions function in terms of (a,b,c) where z is not 1,-1, or 1/2 ? The hypergeometric identities in the mathworld summary...

I find in the literature of the subject: Lorentz transformations and Lorentz-Einstein transformations. The use of one or of the other could lead to a difference in interpretation? Thanks

If a rod is traveling with a velocity 'v' and its proper length is L_0 will the lorentz transformations given below hold true for the length contraction L_0 = \frac {L} { \sqrt {1 - \frac {{v_x}^2} {c^2}}} L_0 = \frac {L} { \sqrt {1 - \frac {{v_y}^2} {c^2}}}

I have studied N.David Mermin "Relativistic addition of velocities directly from the constancy of the velocity of light," Am.J.Phys. 51 1130 1983 and others with the same subject quoted by the Author. He describes a derivation of the addition law that dispenses not only with the LT but also...

Say I have a canonical transformation Q(q,p), P(q,p). In the {q,p} canonical coordinates, the Hamiltonian is H(q,p,t)=p\dot{q}-L(q,\dot{q},t) And the function K(Q,P,t)=H(q(Q,P),p(Q,P),t) plays the role of hamiltonian for the canonical coordinates Q and P in the sense that...

Hi, ok I'm working with linear transformations between normed linear spaces (nls) if T :X -> Y nls's is a linear transformation, we define the norm of T, ||T||: sup{||T|| : ||x||<=1} I want to show that for X not = {0} ||T||: sup{||T|| : ||x|| = 1} frustratingly the...

1) there's given a transformation f:C^n->C^n (C is the complex field), it's known that f is linear on R (real numbers) and its rank on R equals 3 i.e, dim_R Imf=3. now is f linear on C? 2) there's a function f:C->C and its known that f is linear on R, and det_R f<0, is f linear on C? im kind...

Can someone help me with the following? I am supposed to evaluate ∫∫ e^(x+y)dA where the area of integration is given by the inequality |x|+|y|≤1. So, suppose I do one of these Jacobians, and I set u = |x| and v = |y|, so wouldn’t the equation have to satisfy the inequality u+v≤1, and...

I have these past few weeks been steadily studying the different aspects of the theory of Special Relativity. I started with the Lorentz transformations and, thinking I understood them, went along and studied other parts of the theory. However, along the line it has become apparent to me that my...

Please give a plausibile justification for the linearity of the Lorentz transformations. Would you accept: The Lorentz transformations should be linear because to one event in I should correspond a single event in I'

Sorry I don't know how to show the root sign on the forums so I am just going to use / Homework Statement A stretch is applied to the graph of y=/x to produce the graph of y=/9x . Relative to the x and y axis, this stretch may be described as either a... (then it lists a, b, c, or d answers)...

bascially, my teacher rushed us through transformations of functions in an hour, and didnt have time to go throuhg it all so i need some expalnations from you guys please :rolleyes: (you don't have to do it for me, just tell me where to go, thnx) 1.) The curve with equation y = x^2 - 2x -...

I state my problem in the following way. Consider the Lorentz transformations x=g(x’+Vt’) (1) t=g(t’+Vx’/c2). (2) They relate the space-time coordinates of the same events E(x,t) and E’(x’,t’) i.e. the space coordinates of the points M(x,0) and M’(x’,0) where the...

I've been going through "Relativity", a translation of a book by Albert Einstein about the Special and General theories of Relativity. It is stressed that the book should be understandable to anyone with a high school education. "A clear explanation that anyone can understand," it says on the...

Is it correct to say that having the Lorentz-Einstein transformations in our hands we have also all the fundamental equations of special relativity? sine ira et studio

:smile: Discussing with a friend, I was told that using in a derivation the time dilation formula I implicitly use the Lorentz-Einstein transformations.. I mentioned that many Authors derive time dilation without using the LET considering that the transformation equations obscure the physics...

On R^n, I'd say the only smooth transformations taking straight lines to straight lines are the affine transformations. Would I be right saying that?:smile: How would one go about proving that?

Hello, I'll be online until I get this one completely figured out, so baby steps are for the win here. Let L1:U->V and L2:U->W be linear transformations, and let L = L2 * L1 be the mapping defined by: L(u) = L2(L1(u)) for each u which lies in U. Show that L is a linear transformation mapping...

Is the reason why the Lorentz transformations form a group because of the reason on this website http://en.wikipedia.org/wiki/Lorentz_transformation_under_symmetric_configuration So the group consits of 3 matrices, {identity, forward transformations, inverse transformations}?

I understand the concept of relative motion, but I don't know why I can't understand what Galilean Transformations are. Could anyone please explain it? Thank you.

All derivations of the Lorentz Transformations I've seen assume a linear transformation between coordinates. Why must this be the case? Thanks.

Certianly there is a lot of reference material on series transformations: they accelerate convergence, provide analytic continuations and what not. But I have not yet seen a like presentation of product transformations. Given that there are ways to write a product as a series, and vice-versa...

If i am given a linear transformation D:A->A,that is followed by A=ImD(+)kerD and i am asked to prove that kerD^2=kerD and imD=imD^2. instead of trying to work it out the hard way by showing that every element of KerD is an element of kerD^2 , both directions. would it not be easier to...

It is not intuitive, for me at least, why when relating the velocity of 3 inertial frames (Say F1, F2 moving at v1 with respect to F1, and F3 moving at v2 with respect to F2), one mulitplies the transforms of F2 and F3 to get the transform for F1 with respect to F3 to get v3. I understand why v3...

I have a vector (<1, 0, 0>) that needs to be transformed from an initial 3d rectangular coordinate system M1 through M2 and M3 to a final 3d rectangular system M4. I'm currently doing this by applying sequential rotations omega, phi, and kappa about the x', y', and z' axes, respectively, for...

Hello, I've looked through a couple books on this subject and found the basic theory but none actually apply it to a problem. I was wondering if someone would be so kind as to maybe do a practice problem for me? The reason I say this is because I have a homework problem and have...

I have some fairly basic (hopefully) questions about tensor equations. Hopefully someone here can help out. Let us say I have a tensor equation, (I will use this as the example for discussion: A^{u} = b C^{uv}D_{v}). If this is true in one coordinate system, it will be true in all of...

Hi, This is the q. I'm pretty sure that they've got it wrong. Two cars A and B pass each other at a speed of 0.18c. A person in car B says her car is 6.00m long and car A is 6.15m long. What does a person in car A measure for these two lengths? Obviously, the person in car A would...

Hi, I have a question about spinors If \Lambda is a Lorentz Transformation what is (and how do you show that it is) the spinor representation of the Lorentz group ? I think it has somnething to do with the equivalence transformation S\dagger{\gamma}S=\Lambda\gamma But that is just a...

Show that \partial'_{\alpha} A'^\alpha (x') = \partial _{mu} A^{\mu}(x') lets focus on the partial operator for now \partial'_{\alpha} = \frac{\partial}{\partial x'^{\alpha}} = \frac{\partial}{L_{\nu}^{\alpha} \partial x^{\nu}} Now A represents the Scalar and vector fields of an EM...

Givne the Lorentz transformations (LTs)}, x'^{\mu} = L_{\nu}^{\mu} x^{\nu} , between the coordinates, x^{\mu} = (ct , \vec{r}) of an event as seen by O, and coordinates, x'^{\mu} = (ct', \vec{r'}) of the same event as seen by an inertial observer O', show that if we write the inverse...

Let {E1,E2,...En} be an orthogonal basis of Rn. Given k, 1<=k<=n, define Pk: Rn -> Rn by P_{k} (r_{1} E_{1} + ... + r_{n} E_{n}) = r_{k} E_{k}. Show that P_{k} = proj_{U} () where U = span {Ek} well \mbox{proj}_{U} \vec{m}= \sum_{i} \frac{ m \bullet u_{i}}{||u_{i}||^2} \vec{u} right...