Projection Definition and 434 Threads

Homework Statement Hey guys. So here's the situation: Consider the Hilbert space H_{\frac{1}{2}}, which is spanned by the orthonormal kets |j,m_{j}> with j=\frac{1}{2}, m_{j}=(\frac{1}{2},-\frac{1}{2}). Let |+> = |\frac{1}{2}, \frac{1}{2}> and |->=|\frac{1}{2},-\frac{1}{2}>. Define the...

Okay, I thought that there would exist a projection from ##\mathbb{Z}^{p-1}## to ##\mathbb{Z}_{p}## where ##p## is a prime. Like there exist a projection from ##\mathbb{Z}## to ##\mathbb{Z}_p##

Context: T : X \rightarrow X is a measure preserving ergodic transformation of a probability measure space X. Let V_n = \{ g | g \circ T^n = g \} and E = span [ \{g | g \circ T = \lambda g, for some \lambda \} ] be the span of the eigenfunctions of the induced operator T : L^2 \rightarrow...

Hello, Suppose P is a projection operator. How can I show that I+P is inertible and find (I+P)^-1? And is there a phisical meaning to a projection operator? (Please be patient I have just started with QM). Thanks. Y.

Suppose that we have an (n+m)-dimensional tangent space ##T_p^{n+m}## which we decompose into the direct sum of two tangent spaces ##T_p^{n+m} = T_p^n \oplus T_p^m##. We have a coordinate basis in some region of the manifold ##\left\{\partial_{\mu}\right\}_{\mu=1}^{n+m}## from which we want to...

I would like to verify this problem from an introductory to Linear Algebra course. It goes as follows: This is how I proceeded: From the given parametric equations I constructed the vectors: line L: a=(3, -2, 1) and b=(2,1,-2). To find w1, I know that w1= kL And to find k: (v.L)/||L||2 And...

Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V. I'm trying to prove the formula $$R(X',Y'')Z' \cdot V'' = (Z' \cdot (\nabla'_{X'}B') + \left<X'\cdot B', Z' \cdot B'\right>)(Y'', V'') + (V''...

Say we have two vector fields X and Y and we form the projection of Y, Y' orthogonal to X. Since every vector field is associated with a curve with a corresponding parameter, is there a relation between the parameters of Y and Y'?

So I've encountered many "what is the projection of the space curve C onto the xy-plane?" type of problems, but I recently came across a "what is the project of the space curve C onto this specific plane P?" type of question and wasn't sure how to proceed. The internet didn't yield me answers so...

I've noticed that electromagnetic field lines are very similar to stereographic projection of 3D sphere on 2D surface. Pictures below. In such comparison, electric field represents longitude and magnetic field represents latitude. For more visualization see...

My group has given the task of modeling projection of line and my part is to construct the vertical and horizontal plane. So my idea is to have two mirrors joined like an laptop, so that we can fold them and they can also be perpendicular. I'm thinking of making hole in one mirror and a...

Hi there I am trying to project some 3D points on to the span of two orthogonal vectors. v1 = [ -0.1235 -0.9831 0.1352] v2 = [ 0.7332 -0.1822 -0.6552] I used the orthogonal projection formula newpoint = oldpoint-dot(oldpoint,normal(v1,v2))*normal(v1,v2); but when I plot...

Is there already a solution to this available somewhere? Am I not googling it right? I did come upon solutions for this with a point and plane defined by way of vectors and normals but not points. Have I stumbled upon a blank I can fill? Is there room here for a math paper? My approach is...

So, I got a challenge question from my physics teacher but despite attacking it from different angles I’m not quite sure I’ve gotten the correct solution, in fact most of it was just shady assumptions which I’m not sure were supposed to be made, so I’d like to ask whether this is anywhere near...

My problem is that I believe I have a wrong concept somewhere, and I can't find what I'm doing wrong exactly. For this problem let's suppose what I want to do is find the rectangular coordinates of BC. I had two "possible solutions" I tried to achieve this, . First the correct one: (I...

Hello MHB, I have hard to prove that AD, I did put on pic the formula for AD and my progress is at bottom. Regards, |\pi\rangle

Homework Statement A line Ab inclined at 30• to the Hp has its ends a and b, 25 mm and 60 mm behind the vp respectively. The length of the top view is 65 mm and its vt is 15 mm below the Hp. Draw the projections of the line and locate its ht. also, determine the true length of the line Ab and...

Hi, Consider you are standing upright and pointing your finger at the ground. Where does the vector coming off the tip of your finger arrive when it hits ground level on the other side of the Earth? ..Think as if you were going to imperviously dig a hole through the Earth and could travel...

Homework Statement (i) Find the normal, n, at a general point on the surface S1 given by; x2+y2+z = 1 and z > 0. (ii) Use n to relate the size dS of the area element at a point on the surface S1 to its projection dxdy in the xy-plane. The Attempt at a Solution To...

I am trying to solve the following problem on an old Quantum Mechanics exam as an exercise. Homework Statement Homework Equations I know that the trace of an operator is the integral of its kernel. \begin{equation} Tr[K(x,y)] = \int K(x,x) dx \end{equation} The Attempt at a...

Homework Statement The angle between two vectors a and b is ∅, where ∅≠90°. Under what conditions will |Projab|2 + |Projba|2 = 1? Homework Equations |Pab|= |\vec{a}\bullet\vec{b}|/|\vec{b}| |Pba|= |\vec{a}\bullet\vec{b}|/|\vec{a}| The Attempt at a Solution I know that...

Hello, I was wondering, why is the vector projection useful in the way that it is presented? Why isn't just the vector times cosθ sufficient to find the projection of a vector onto another one, why the dot product divided by the magnitude of the vector squared times that same vector? The...

I have bumped into a term a^\dagger \hat{O}_S | \psi \rangle I would really like to operate on the slater determinant \psi directly, but I fear I cannot. Is there any easy manipulation I can perform?

Verify the spectral thrm for the symmetric matrix, by finding an orthonormal basis of the apporpriate vector space, the change of basis matrix to this basis and the spectral decmoposition. Well I've found everything else. We started with the matrix A. A = \begin{bmatrix} 2 & 3 \\ 3 & 2...

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...

I cannot quite understand why expression \frac{1-\gamma_5 \slashed{s}}{2} is covariant? We defined it in the rest frame, and then said that because it is in the slashed expression, it's covariant, what does that mean? s is the direction of polarization, s \cdot s = -1

Homework Statement Hello, H is a Hilbert space. K is a nonempty, convex, closed subset of H. Prove that the orthogonal projection Pk: H → H, is non-expansive: ll Pk(x) - Pk(y) ll ≤ ll x - y ll The Attempt at a Solution So the length between the Pk's, which is in K (convex) is less than...

I am recently trying to identify system parameters with projection algorithm, but faced a problem, and the dynamic model is the following: \ddot{y}(t)+a\cdot\dot{y}(t)=b\cdot e(t) The true value of a is 2.8, b is 0.1. While inputing volt e(t)=12sin(2\pi t)+5sin(2t), I can get a good...

Homework Statement Let \vec{X(t)}: I \rightarrow ℝ3 be a parametrized curve, and let I \ni t be a fixed point where k(t) \neq 0. Define π: ℝ3 \rightarrow ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to \vec{X(t)} at t. Define γ=π\circ\vec{X(t)} as the orthogonal projection...

Homework Statement For a given velocity of projection in a projectile motion, the maximum horizontal distance is possible only at ө = 45°. Substantiate your answer with mathematical support. Homework Equations My teacher gave us the information that u=10 m/sec, however I don't see...

Homework Statement Let S be the linear span of the orthogonal set: {[3 2 2 2 2]T,[2 3 -2 -2 -2]T,[2 -2 3 -2 -2]T} Calculate the orthogonal projection of Y = [1 2 -1 3 1]T onto S. The Attempt at a Solution Not sure how to go about this... Do i find a vector that is orthogonal...

Simple question, but I don't know why I never learned this before. If the scalar projection of vector B onto vector A is B * Unit vector of A (or [A dot B]/[magnitude of A]), then what does the dot product of simply A and B give you, assuming neither is a unit vector. If it's not clear what...

Homework Statement Let v be a non-zero (column) vector in Rn. (a) Find an explicit formula for the matrix Pv corresponding to the projection of Rn to the orthogonal complement of the one-dimensional subspace spanned by v. (b) What are the eigenvalues and eigenvectors of Pv? Compute the...

Is the following statement true? My intuition tells me it is true, but I have been trying to prove it, without much success: (proj_{v}u ) \cdot (u - proj_{v}u) = 0 It makes complete since if you draw it out in R2, but I am trying to prove it in Rn. Any ideas? BiP

Homework Statement The turnbuckle is tightened until the tension in the cable AB equals 2.9 kN. Determine the vector expression for the tension T as a force acting on member AD. Also find the magnitude of the projection of T along the line AC. Homework Equations The Attempt at...

Homework Statement Express the 5.2-kN force F as a vector in terms of the unit vectors i, j, and k. Determine the scalar projections of F onto the x-axis and onto the line OA. I have attached an image of the problem. Homework Equations Fx = Fcos(θ) Fy = Fcos(θ) Fz = Fcos(θ)...

We know stereographic projection is conformal but it isn't isometic and in general relativity it can not be used because in this theory general transformations must be isometric. But de sitter in his model (1917) used it (stereographic projection) to obtain metric in static coordinates. How...

Hi guys, my first post here. This isn't homework or anything, just a personal project. Homework Statement I need to find the inverse of the relative mercator equation. That is, given an origin latitude/longitude, find the current latitude/longitude from an x,y point. Because longitude is...

So here's the situation, say i record, presuming the camera is perfectly vertical, an object falling next to a ruler. I then find how many pixels that ruler is and from there i can translate how many pixels the object has moved into how far the object has moved. The question is how reliable is...

Hello, I am looking at some code which creates a projection matrix and I can verify that it is indeed correct as P^2 = P. The way they do is as follows: There is a 4x4 matrix which is an affine map between two coordinate systems (takes one from image space to world space). It is a...

Hi everyone! Which is the formula and the proof of the projection of the angular momentum of a rigid body along the rotation axis? I searched on the web and on my mechanics book but cannot find anything... does somebody know this curiosity ?

Hi, I'm trying to create a projection on a flat surface by using an LCD screen extracted from a flat panel monitor, a sheet of translucent fine grain paper, a Fresnel lens and an LED light source. The purpose is to project a focused image on the paper for display proposes. The idea is to...

Homework Statement Let T : M2x2(ℝ) → M2x2(ℝ) denote the projection on W along U. W={ [a b] [c d] : a+b+c+d=0 } U= span{ I2 } Find the matrix representation of T with respect to the standard basis of M2x2(ℝ) and the formula for T. Homework Equations From my notes, I know W is...

Homework Statement Let S be the subspace of R3 defined by x1 - x2 + x3 = 0. If L: R3 -> R3 is an orthogonal projection onto S, what are the eigenvalues and eigenspaces of L? Homework Equations The Attempt at a Solution First off, I hadn't seen the term eigenspace before. From...

Homework Statement A person is standing 80 ft from a tall cliff. She throws a rock at 80 ft/sec at an angle of 45 degrees from the horizontal. Neglecting air resistance and discounting the height of the person, how far up the cliff does it hit? Homework Equations r = ((xo + vo*cos(~))t) i...

Homework Statement In each case describe the eigenvalues of the linear operator and a base in R^3 that consist of eigenvectors of the given linear operator. Write the matrix of the operator with respect to the given base. The Orthogonal Projection on the plane 2x + y = 0 and...

Stumped on #15. I feel like its much easier than I am making it out to be and that maybe I am just over thinking. Any help or leads would be appreciated. Thanks for your time...

Hi! Newly registered here, with an optics question. I was going through some photos I took of the transit of Venus this year, where I'd used a reflecting telescope to project the image of the sun and the passing planet onto a piece of white paper. Image: http://imgur.com/3mdUN I showed it...

Dear all, I want to prove that a circular hodograph (planetary orbit in momentum 3-space) stereographically projects onto a great circle of a 3-sphere in momentum 4-space. The equation for the hodograph is given by: \begin{equation} \left( \frac{mk}{L} \right)^2 = p_1'^2 + \left( p_2' -...

The title says it. I would like to see what knowledgeable people at PF have to say about the QM projection postulate -- primarily understandings of the conceptual reasoning underlying it. But anything anyone has to say about it is welcomed, including opinions that it shouldn't be a part of the...