Orthogonal Definition and 560 Threads

what exactly are orthogonal matrices? can someone give me an example of how they would look like?

Let A= [ 1 -2 -1 2] [-1 0 3 -2 ] [ 3 -4 -5 6] (sorry, can't line up the columns) I think I've done a) and b) correctly, I don't really understand c), d) and e) a) Find RowA and Nul A RowA={( 1, -2, -1, 2), (0, 1, -1, 0)} Nul A= {(3, 1, 1, 0), (-2, 0, 0, 1)} b) If...

Homework Statement I have a general question. If we have some subspace W of R^n where dimW=k. Then if T is an orthogonal transformation from R^n->R^n is the dimension of T(W) also k? Homework Equations The Attempt at a Solution The reason I think this is true is because if...

Find an orthogonal transformation T from R3 to R3 such that T of the column vector [2/3 2/3 1/3] is equal to the column vector [0 0 1] So I tried to construct out the 3x3 matrix [a b c] [d e f] [g h i] and applied the properties of an orthogonal matrix and basic algebra. I ended up with a...

what is the meaning of orthogonal relationships in addition to right angles in the xyz coordinate system? for instance, if a 3 piece rocket separated in space in an orthogonal way...will there be any significance when compared to the 3 piece rocket that does not separate in an orthogonal...

Given: \vec A \cdot \vec B = non zero and \theta does not equal 0 I can't seem to prove that Vector B minus the Projection of B onto A makes the orthogonal projection of B onto A. Can you help?

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...

We know that if M is an orthogonal matrix,then DetM=(+-)1 When Det M=1,thee transformation is a rotation.And for reflection about anyone o all three axes DetM=-1. I did this.. But I did not know that information:When Det M=1,thee transformation is a rotation.And for reflection about anyone...

Homework Statement For which real numbers c and d does the matrix have real eigenvalues and three orthogonal eigenvectors? 120 2dc 053 Homework Equations im having trouble getting started on this one. Ive tried using solving for the eigenvalues pretending that c and d are...

In my intro to E&M course, in the section on electromagnetic waves, my textbook just says that electric and magnetic waves are orthogonal to each other, but it doesn't say why. How do we know this? Is it from solving the wave partial differential equation? If so, given that I've tooken a course...

I'm a bit confused, conceptually. This is the problem Let v1=( 1, -1, 2) v2=( 2, 1, 3) v3=( 1, -4, 3) Find a nonzero vector u that is orthogonal to all three vectors v1, v2, and v3. I know how to find the projection matrix, P, which I can solve with v1, v2, and v3. The equation for that is...

Homework Statement Consider a symmetric n x n matrix A with A² = A. Is the linear transformation T(x) = Ax necessarily the orthogonal projection onto a subspace IR^n? Homework Equations The Attempt at a Solution No idea what thought to begin with.

The question is (true or false) if Q is an improper 3 x 3 orthogonal matrix then Q^2 = I. The way I have approached it so far has been a brute force method. I'm not really sure if this will be true or false, and I have a feeling it is false, but I can't construct a good counter-example...

If two vectors v, w are both unit vectors, then v+w and v-w will be orthogonal, but are v+w and v-w also unit vectors? I would say no because the inner product of the two added, and the two subtracted would also have to be orthogonal. <(v+w)+(v-w),(v+w)-(v-w)> = <2v,2w> and <2v,2w> must be...

Why is it that eigenfunctions of different excited states for 1 atom have to be orthogonal?

1. Homework Statement . The correct answer is E 2. Homework Equations :Procedure from our text: "Step 1. Determine the differential equation for the given family F(x, y,C) = 0. Step 2. Replace y' in that equation by −1/y'; the resulting equation is the differential equation for the family of...

Homework Statement Write the equation to find the orthogonal trojectory for x^2 + y^2 = r^2 2. The attempt at a solution Well.. you solve this by just integrating.. but, I don't know what to do with the r^2 ? x^2 + y^2 = r^2 x^2 + y^2 \frac{dy}{dx} = r^2 \frac{dy}{dx} is that...

Homework Statement Show that if the vector \textbf{v}_1 is a unit vector (presumably in \Re^n) then we can find an orthogonal matrix \textit{A} that has as its first column the vector \textbf{v}_1. The Attempt at a Solution This seems to be trivially easy. Suppose we have a basis \beta for...

Assume that I is the 3\times 3 identity matrix and a is a non-zero column vector with 3 components. Show that: I - \frac{2}{| a |^{2}}aa^{T} is an orthogonal matrix?My question is how can one take the determinant of a if it is not a square matrix? Is there a flaw in this problem? Thanks

We were given a graded assigment and one of the question asks. Prove that all curves in the family y1=-.5x^2 + k (k any constant) are perpendicular to all curves in the family y2=lnx+ c (c any constant) at their points of intersection. I found the derivatives of y1 and y2 and they are...

A question for my understanding: If I have an operator \cal{L} and a set of eigenfunctions \phi_n of this operator, then the eigenfunctions are orthogonal. Why is that?

In the first hald of this question it was proven that -\frac{\hbar^2}{2m} \frac{d}{dx} \left[ \phi_{m}^* \frac{d \phi_{n}}{dx} - \phi_{n} \frac{d \phi_{m}^*}{dx}\right] = (E_{m} - E_{n}) \phi_{m}^* \phi_{n} By integrating over x and by assuming taht Phi n and Phi m are zero are x = +/-...

I have to find the orthogonal trajectory to the family of circles tangents to the x-axis. I eventually found that the derivative of the orthogonal trajectory would be: \frac{dy}{dx}= \frac{-x^2 + y^2}{2xy} But how do i find the equation for "y" explicitly? (ps. This is not an assignment...

Hi all! This looks a pretty nice forum. So here's my question: How do I find/show a basis or orthogonal basis relative to an inner product? The reason I ask, is because in my mind I see the inner product as a scalar, and thus I find it difficult to "imagine" how a scalar lives in a...

A general orthogonal coordinate system (u,v,w) will have a line elemet of the form: ds^2 = f^2 du^2 + g^2dv^2 + h^2dw^2 I have done a lot of vector calculus, but for some reason I can't figure out what this means! What is a line element? I know about the differential length element and its...

I'm doing a series of questions right now that is basically dealing with the dot and cross products of the basis vectors for cartesian, cylindrical, and spherical coordinate systems. I am stuck on \hat R \cdot \hat r right now. I'll try to explain my work, and the problem I am running into...

I am working on this problem, and have a simple question. Determine the orthogonal trajectory of x^p + Cy^p = 1 where p = constant. I start out by taking the derivative with respect to x. My question is this. does Cy^p become Cpy^{p-1} or C_1y^{p-1} ? Thanks, Morgan

I've been banging my head against this problem for some time now, and I just can't solve it. The problem seems fairly simple, but for some reason I don't get it. Given the coordinate transformation matrix A=\left(...

After 10 years of teaching middle school, I am going back to grad school in math. I haven't seen Linear Algebra in more than a decade, but my first class is on Generalized Inverses of Matrices (what am I thinking?). I have a general "rememberance" understanding of most of the concepts we're...

I need some help in understanding what I need to do to solve these poblems, I can't get them started. 1. Find an orthogonal set of vectos that spans the same subspace as a,b,c. a=(1,1,-1) b=(-2,-3,1) c=(-1,-2,0) 2. Use the Gram-Schmidt process to find and orthogonal basis that...

I have the set s = span ( [[0][1][-1][1]]^{T} ) And I need to find the orthogonal complement of the set. It seems like it should be straight foward, but I'm a bit confused. I know that S is a subspace of R^4, and that there should be three free vairables. What I did so far is to take the...

I've got a question regarding orthogonal matrices. I am given an orthogonal matrix M, and a symmetric matrix A. I need to prove that (M^-1)*A*M is also symmetric (all of the matrices are n x n). I know that for an orthogonal matrix, its inverse is equal to its transpose. Can anyone give me...

I have a question on matrix norms and orthogonal transformations. The 2-norm in invariant under orthogonal transformation, for if Q^T*Q=I. But i have trouble showing that for orthogonal Q and Q^H with appropriate dimensions || Q^H*A*Q ||2 =|| A ||2

Plz Help :( Hi I want 2 know how 2 solve 1st order partial differintial equation (PDE) with constant coefficient using orthogonal transformation example : solve: 2Ux + 2Uy + Uz = 0 THnx :blushing:

Hello, I have a "simple" problem for you guys. I am not expert in math and so try to be simple. I explain the problem by starting with one example. The polar coordinate system has the following main property: with two parameters, rho and theta, each point is described as the intersection of...

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...

Let T in L(V) be an idempotent linear operator on a finite dimensional inner product space. What does it mean for T to be "the orthogonal projection onto its image"?

Show that the functions sin x, sin 2x, sin 3x, ... are orthogonal on the interval (0,pi) with respect to p(x) = 1 (where p is supposed to be rho) i know i have to use this \int_{0}^{\pi} \phi (x) \ psi (x) \rho (x) dx = 0 and i have no trouble doing it for n = 1, and n=2 but how wouldi go...

Hello again, This question confuses me for a reason. I read the questions and they sound to simple and to easy to answer. So maybe its something I am reading wrong and not answering. Help would be greatly apreciated. first off Let O(n) = { A | A is an n x n matrix with A^t A = I } be the...

Find an equation of a plane through the point (-1, -2, -3) which is orthogonal to the line x=5+2t,y=-3-5t,z=2-2t in which the coefficient of x is 2. ______________________________=0 i don't get this problem at all, but here's what i came up with after sitting here at the computer for a...

Find a vector orthogonal to both <-3,2,0> and to <0,2,2> of the form <1,_,_> (suppose to fill in the blanks) well i thought the cross product would do the trick, but i keep getting the wrong answer. I|2 0| - j |-3 0| + k |-3 2| |2 2| |0 2| |0 2| (format is kinda messed up...

i am still learning mathematical physics. i am learning orthogonal polynomials, but still confused. what is the meaning of "orthogonal" here?

Show that x+y and x-y are orthogonal if and only if x and y have the same norms. Can someone get me started?

What is the difference between these terms? In what context do they apply to? How important is it that we treat them differently?

Hi everybody, I read that sin(2x) and sin(3x) are orthogonal to each other. In general if I want to check if two functions are orthogonal or not I must integrate their product First: why the integration of their multiplication (not their addition for example)? Second: Orthogonal...

find the orthogonal trajectories of the following (a) x^2y=c_1 (b) x^2+c_{1}y^3=1 for part (a) I've found y=\frac{1}{2}\log{|x|} + C_2 for part (b) if i solve this integral this should be the O.T. \frac{3}{2}\int{(\frac{1}{x^2}-1)}dx= \frac{y^2}{2} is this correct?

A thm says: if W is a subspace of V then V = direct sum of W and CW( ort. complement of W) i.e. for all v € V there exist w € W & w' € CW s.t. v= w+w' Does it mean that we can write a function as a sum of two orthogonal funcs ? Also i don't know the proof...

How can you prove that the set of orthogonal matrices are compact? I know why they are bounded but do not know why they are closed.

Is it correct to assume that there is no such thing as non-orthogonal basis? The orthogonal eigenbasis is the "easiest" to work with, but generally to be a basis a set of vectors has to be lin. indep and span the space, and being "lin. indep." means orthogonal. Is it correct? Thanks.

How do I prove that two curves are orthogonal when they interest each other at a specific point? Do I just take the derivative of both and compare the slopes? The slopes should be negative reciprocals of each other, correct?