Orthogonal Definition and 560 Threads

Homework Statement Let P\inL(V). If P^2=P, and llPvll<=llvll, prove that P is an orthogonal projection.Homework Equations The Attempt at a Solution I think that regarding llPvll<=llvll is redundant. For example, consider P^2=P and let v be a vector in V. Doesn't P^2=P kind of give it away by...

Aren't all projections orthogonal projections? What I mean is that let's say there is a vector in 3d space and it gets projected to 2d space. So [1 2 3]--->[1 2 0] Within the null space is [0 0 3], which is perpendicular to every vector in the x-y plane, not to mention the inner product of...

I know what orthogonal means (well, I know orthogonal vectors are perpendicular to each other) but how can this be applied to a wavefunction? Thanks!

Homework Statement Let W be the subspace spanned by the given column vectors. Find a basis for W perp. w1= [2 -1 6 3] w2 = [-1 2 -3 -2] w3 = [2 5 6 1] (these should actually be written as column vectors. Homework Equations The Attempt at a Solution So, I...

Hi, I was just reading about Orthogonal complements. I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V. I then proved that the intersection of W and its orthogonal complement equals 0...

Homework Statement I would like to show that SO3 does not contain any subgroups that are isomorphic to SO2 X SO2. Homework Equations I know that any finite subgroup of SO3 must be isomorphic to a cyclic group, a dihedral group, or the group of rotational symmetries of the tetrahedron...

I need some direction. Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.

I need some direction. I don't have a clue where to start. Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.

Show that the parabolic coordinates (u,v,\phi) defined by x=uv \cos{\phi} , y=uv \sin{\phi} , z=\frac{1}{2}(u^2-v^2) now I am a bit uneasy here because to do this i first need to find the basis vector right? so if i try and rearrange for u say and then normalise to 1 that will give me...

Homework Statement Let P2 denote the space of polynomials in k[x] and degree < or = 2. Let f, g exist in P2 such that f(x) = a2x^2 + a1x + a0 g(x) = b2x^2 + b1x + b0 Define <f, g> = a0b0 + a1b1 + a2b2 Let f1, f2, f3, f4 be given as below f1 = x^2 + 3 f2 = 1 - x f3 = 2x^2 + x + 1 f4 = x +...

Homework Statement Find an orthogonal basis for the nullspace of the matrix [2 -2 14] [0 3 -7] [0 0 2] Homework Equations The Attempt at a Solution The nullspace is x = [0, 0, 0], so what is the orthogonal basis? It can be anything can't it?

Given a = ( 1, -2, 1), b= ( 0, 1, 2), and c = (-5, -2, 1) determine if {a, b, c} is an orthogonal set. Show support for your answer. I know that if the dot product of every combination equals zero, the set is orthogonal. No problems here. I do that, and they all equal zero. a dot b: 1 * 0 +...

I have an assignment question to find an equation of the orthogonal projection onto the XY plane of the curve of intersection of twp particular functions. If some one knows of a good web page that might explain this to me I would be greatly appreciate it. regards Brendan

V is a space of inner muliplication. W1 and W2 are two subspaces of V, so dimW1<dimW2 prove that there is a vector 0\neq v\epsilon W_2 which is orthogonal to W1 ??

W is sub space of R^4 which is defined as http://img21.imageshack.us/img21/1849/63042233.th.gif find the system that defines the complements W^\perp of W i have solved the given system and i got one vector (-1,1,0,0) so its complement must be of R^3 and each one of the complements...

Let O(n,F_q) be the orthogonal group over finite field F_q. The question is how to calculate the order of the group. The answer is given in http://en.wikipedia.org/wiki/Orthogonal_group#Over_finite_fields". This seems to be a standard result, but I could not find a proof for this in the basic...

Imagine there is a molecule which consists of several atoms, and for each atom there is an effective orbital, phi_i, which are not orthogonal. Now we want to construct from them a set of orthogonal orbitals, psi_i. Of course there are many ways to do this. Let W be the matrix that realizes our...

Homework Statement Find all vectors that are perpendicular to (1,4,4,1) and (2,9,8,2) The Attempt at a Solution Create matrix A = [[1,4,4,1],[2,9,8,2]] Set Ax = 0 Reduce by Gauss elimination Produces basis of (-4,0,1,0) and (-1,0,0,1) I don't know what the correct solution to...

I'm wondering about the action of the complex (special) orthogonal group on \mathbb{C}^n. In the real case, we can use a (real) orthogonal matrix to rotate any (real) vector into some standard vector, say (a,0,0,...,0), where a>0 is equal to the norm of the vector. In other words, the action...

Homework Statement If w is orthogonal to u and v, then show that w is also orthogonal to span ( u , v ) Homework Equations two orthogonal vectors have a dot product equalling zero The Attempt at a Solution I can see this geometrically in my mind, and I know that w . u = 0 and w ...

Homework Statement 1. If I got a square orthogonal matrix, then if I make up a new matrix from that by rearranging its rows, then will it also be orthogonal? 2. True/false: a square matrix is orthogonal if and only if its determinant is equal to + or - 1 Homework Equations no...

Homework Statement I don't understand the difference between an orthogonal complement and it's basis. In this problem: W = [x,y,z]: 2x-y+3z=0 Find w's orthogonal complement and the basis for the orthogonal complement. The Attempt at a Solution I did a quick reduced row echelon to [2,-1,3]...

Homework Statement Prove that an orthogonal transformation T in Rm has 1 as an eigenvalue if the determinant of T equals 1 and m is odd. What can you say if m is even? The attempt at a solution I know that I can write Rm as the direct sum of irreducible invariant subspaces W1, W2, ..., Ws...

Greetings, I'm a little stuck on this algebra question Find 2 unit vectors orthogonal to v1 = <3,1,1,-1>, v2 = <-1,2,2,0> and v3 = <1,0,2,-1> I know that this means that if I let z be the vector orthogonal to these 3 then, z.v1 = z.v2= z.v3 = 0 And that my two vectors is likely +/-...

Homework Statement http://img55.imageshack.us/img55/8494/67023925dy7.png The Attempt at a Solution All functions orthogonal to 1 result in the fact that: \int_a^b f(t)\ \mbox{d}t =0 Now the extra condition is that f must be continous. (because of the intersection). But...

Homework Statement Consider the vector space \Renxn over \Re, let S denote the subspace of symmetric matrices, and R denote the subspace of skew-symmetric matrices. For matrices X,Y\in\Renxn define their inner product by <X,Y>=Tr(XTY). Show that, with respect to this inner product, R=S\bot...

I'm sure there's a simple way of doing this but I just can't think of it. In R^n, given k<n linearly independent vectors, how do you find a vector that is orthogonal to all given vectors. I know cross product works for R^3, but what about R^n?

Homework Statement Given vectors a=(1,-1,0,2,1) b=(3,1,-2,-1,0) and c=(1,5,2,4,-4), which are mutually orthogonal, find a system of linear equations that a vector x must satisfy so it is orthogonal to a, b, and c.Homework Equations None I think. The Attempt at a Solution This is part A of...

hi! I'm new to the forums, and had a question that was more calculus-related than physics. i saw another post similar to this one, but it was incomplete and i couldn't get the answer with the information on it, any chance someone could help me out? The question is: "Find a unit vector with a...

Homework Statement The problem is more of a concept/visual problem, like to find out if it's true or false and why: all vectors orthogonal to a non zero vector in R^3 are contained in a straight line. Homework Equations No equations, just very bad at visualizing in 3-D The...

I have a test tomorrow and the book does not even explain it that well. Than You

How can I get the central force law by using orthogonal transformation of position vector, x=Ar where A is an orthogonal matrix and r is a position vector? Thanks!

a) Let M be a 3 by 3 orthogonal matrix and let det(M)=1. Show that M has 1 as an eigenvalue. Hint: prove that det(M-I)=0. I think I'm supposed to begin from the fact that det(M)=1=det(I)=det(MTM) and from there reach det(M-I)=0 which of course would mean that there's an eigenvalue of 1 as...

Homework Statement Let W be the subspace of R^3 spanned by the vectors v1=[2,1,-2] and v2=[4,0,1] Find a basis for the orthogonal complement of W Homework Equations None The Attempt at a Solution I can do this question except for the fact when i get the matrix in form...

Hi everyone, I was reading about Gram-Schmidt process of converting two linearly independent vectors into orthogonal basis. But, as I understand, if two vectors are linearly independent then they are orthogonal! isn't that right?? Could anybody explain... please.

Homework Statement Show that the product of two Orthogonal Matrices of same size is an Orthogonal matrix. I am a little lost as to how to start this one. If A is some nxn orthogonal matrix, than AA^T=A^TA=I where I is the nxn identity matrix. So now what? Let's take A and B to be nxn...

On page 102 of Introduction to Quantum Mechanics, Griffiths writes that \int_{-\infty}^{\infty}e^{i\lambda x}e^{-i \mu x}dx = 2\pi\delta(\lambda-\mu) I don't see how this follows. If you replace lambda with mu, then you get a cancellation, and the integral of 1dx. Am I missing something?

Suppose A\overrightarrow{x}=\lambda_1\overrightarrow{x} A\overrightarrow{y}=\lambda_2\overrightarrow{y} A=A^T Take dot products of the first equation with \overrightarrow{y} and second with \overrightarrow{x} ME 1) (A\overrightarrow{x})\cdot...

Homework Statement My book used the term "orthogonal 3 by 3 matrix" and I couldn't find where that was defined. Does that mean that the column vectors are orthogonal or that the row vectors are orthogonal? Or are those two things equivalent? Homework Equations The Attempt at a Solution

[SOLVED] Quick Question: Is this matrix an orthogonal projection? Homework Statement P=[0 0 ] [11] Homework Equations The Attempt at a Solution Its orthogonal if the null space and range are perpendicular. Range=[0 ] [x+y] null space=[x

Homework Statement Prove that if P in L(V) is such that P2 = P and every vector in Ker(P) is orthogonal to every vector in Im(P), then P is an orthogonal Projection. Homework Equations Orthogonal projections have the following properties: 1) Im(P) = U 2) Ker(P) = Uperp 3) v - P(v)...

Homework Statement My statistical mechanics book says that if M is an real, symmetric n by n matrix, and U is the matrix of its eigenvectors as column vectors, then the transformation UMU^{-1} preserves the trace of M. Is that true? If so, is it obvious? If it is true but not obvious, how do...

Homework Statement Orthogonally diagonalize the matrix: | 2 1 1| | 1 2 1| | 1 1 2| Homework Equations Since this only has...

Homework Statement Determine whether the given vectors are orthogonal, parallel, or neither. Homework Equations \cos \theta = \frac{{\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} }}{{\left| {\overrightarrow {\rm{a}} } \right|\left| {\overrightarrow {\rm{b}} } \right|}}\,\...

[SOLVED] Hilbert space &amp; orthogonal projection Homework Statement Let H be a real Hilbert space, C a closed convex non void subset of H, and a: H x H-->R be a continuous coercive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M \geq 0 such that |a(x,y)| \leq...

Hi I had a final today and one of the questions was find all the possible values of det Q if Q is a orthogonal matrix I m still wondering how would I do this? Any ideas?

[SOLVED] orthogonal eigenfunctions From Sturm-Louisville eigenvalue theory we know that eigenfunctions corresponding to different eigenvalues are orthogonal. For example, \Phi_{xx} + \lambda \Phi = 0 would be of Sturm-Louisville form (note: \Phi_{xx} represents the second derivative of...

Homework Statement Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y = x/(1+kx) 2. The attempt at a solution I have been trying this problem for hours, and I get a different answer every time...

Homework Statement Two curves intersect orthogonally when their tangent lines at each point of intersection are perpendicular. Suppose C is a positive number. The curves y=Cx^2 and y=(1/x^2) intersect twice. Find C so that the curves intersect orthogonally. For that value of C, sketch both...

Let the function f(z) = u(x,y) + iv(x,y) be analytic in D, and consider the families of level curves u(x.y)=c1 and v(x,y)=c2 where c1 and c2 are arbitrary constants. Prove that these families are orthogonal. More precisely, show that if zo=(xo,yo) (o is a subscript) is a point in D which is...