Basis Definition and 1000 Threads

I've just studied the implicit function theorem and if we assume the theorem is true then we can easily compute the following: id_n = D(id_n)_x = D({f^-1}°{f})_x = {D(f^-1)_f(x)} ° {Df_x} where D(*)_a means the derivative of * at x. OKay... so this was very straightforward until I began...

Hi There, I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a...

Homework Statement Is the system of vectors a1=(1,-1,0,1), a2=(2,3,-1,0), a3=(4,1,-1,4) linearly independent? Do these vectors form a basis in the vector space R^4? State why. Homework Equations The Attempt at a Solution I have done the first part of the exercise. I have found...

Homework Statement Let A be an 3x3 matrix so that A^3 = {3x3 zero matrix}. Assume there is a vector v with [A^2][v] ≠ {zero vector}. (a) Prove that B = {v; Av; [A^2]v} is a basis. (b) Let T be the linear transformation represented by A in the stan- dard basis. What is [T]B? Homework...

Normally, you need how system transforms n basis vectors to say how it transforms arbitrary vector. For instance, when your signal is presented in Fourier basis, you need to know how system responds to every sine. But, I have noted that it is not true for the simplest standard basis. You just...

First of all, I'd like to say hi to all the peole here on the forum! Now to my question: When reading some general relativity articles, I came upon this strange notation: T^{a}_{b} = C(dt)^{a}(∂_{t})_{b} + D(∂_{t})^{a}(dt)_{b}. Can someone please explain to me what this means? Clearly...

Hi there, I am reading the book "Condensed Matter Physics" second edition by Michael P. Marder. It stated in page 9 that one basis of the the honeycomb lattice is \vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad \vec{v}_2 = a [0 \ -1/(2\sqrt{3})] which is based on figure 1.5(B) in page...

Find a basis and the dimension for the subspace of R^3(3D) spanned by the vectors {(0,1,-2),(3,0,1),(3,2,-3)} The dimension is 2 regardless if i put the vectors in row space or column space form. But to find the basis I need to put it in row space form. Can anyone please explain when I...

The set is (1,3),(-1,2),(7,6) it is in R2 so I don't get why there are 3 elements. I assumed they are not vectors but points instead. but if they are points then it becomes a line, and the answer is that its dimension is 2, and a basis is (1,0) and (0,1) Could someone explain this? Thanks

http://dl.dropbox.com/u/33103477/linear%20transformations.png My solution(Ignore part (a), this part (b) only) http://dl.dropbox.com/u/33103477/1.jpg http://dl.dropbox.com/u/33103477/2.jpg So I have worked out the basis and for the kernel of L1 and image of L2, so I have U1 and U2...

I am working on a problem dealing with transformations of a vector and finding the basis of its kernel. Now I have worked out everything below but after reading the definitions I am a bit confused, hence just want verification if the procedure I am following is correct. My transformed matrix is...

http://dl.dropbox.com/u/33103477/linear%20transformations.png My attempt was to first find the transformed matrices L1 and L2. L1= ---[3 1 2 -1] -------[2 4 1 -1] L2= ---[1 -1] -------[1 -3] -------[2 -8] -------[3 -27] Now reducing L1, I have -------[1 0 7/10 -3/10]...

Hi everyone, Pardon the neophyte question, but is a one-form the same thing as a dual basis vector? If not, are they related in some way, or completely different concepts/entities? Thank you!

Can somebody help me how to approach this problem.I am having trouble finding the orthogonal basis.

B&F have the following: \delta_{i j} = e'_i \cdot e'_j = a_{i k} \left( e_k \cdot e'_j \right) = a_{i k} a_{j k} and they ask the reader to show that a_{k i} a_{k j} = \delta_{i j} Does it suffice to show the following? : \delta_{i j} = a_{i k} a_{j k} \to \delta_{j i} = \left(...

How the hell do you prove that the components of a vector w.r.t. a given basis are unique? I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!

hello :) I was trying to prove the following result : for a linear mapping L: V --> W dimension of a domain V = dimension of I am (L) + dimension of kernel (L) So, my doubt actually is that do we really need a separate basis for the kernel ? Theoretically, the kernel is a subspace of the...

Homework Statement Let W be a subspace of ℝ4 spanned by the vectors: u1 = [1; -4; 0; 1], u2 = [7; -7; -4; 1] Find an orthogonal basis for W by performing the Gram Schmidt proces to there vectors. Find a basis for W perp (W with the upside down T). Homework Equations Gram...

First off, I've never taken a differential equations class. This is for my Math Methods for Physicists class, and we are on the topic of DE. Unfortunately, we didn't cover this much, so most of what I am about to show you comes from the professor giving me tips and my own common sense. I'd...

Homework Statement Let s be the linear transformation s: P2→ R^3 ( P2 is polynomial of degree 2 or less) a+bx→(a,b,a+b) find the matrix of s and the matrix of tos with respect to the standard basis for the domain P2 and the standard basis for the codomain R^3 The Attempt at a...

Homework Statement I'm trying to understand the preferred basis problem in the foundations of QM Ok so I read somewhere that in general any state can be decomposed in different ways. I don't quite see how this is meant to work Suppose 'up' / 'down' represent z component of ang mom...

Homework Statement This is the question on my assignment: In each case below, given a vector space V , find a basis B for V containing the linearly independent set S ⊂ B. It has a bunch of different cases but I think that if you help me with the following two, I will learn enough to do...

Homework Statement The Attempt at a Solution So first I thought to myself that the proper way of doing this problem was to construct each of the standard basis vectors as a linear combination of the basis given us. I have, T(1,0,0) = \frac{1}{2} T(1,0,1) + \frac{1}{2} T(1,0,-1) =...

Homework Statement Let V = P_n(\textbf{F}) . Prove the differential operator D is nilpotent and find a Jordan basis. Homework Equations D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} The Attempt at a Solution I already did the proof of D being nilpotent, which was easy. But we haven't covered...

I have S= {(1,1,0,1) (1,0,-1,0) (1,1,0,2)} its one of the subset and second it T= {(x,y,z,2x-y+3z)} If you were to use Gram-Schmidt method to find the orthogoan basis for T who would you processed? I really don't understand this concept. I know from T , the hyperplane is 2x-y+3z so the...

Does anyone know if a set of homogeneous polynomials forms a reduced Grobner basis, then they form a regular sequence in the polynomial ring? Any references? All the references that I have looked at (so far) have not related the two. If this is not true, can you give me a counterexample...

Hi, I have some very basic questions regarding electron energy levels/states. In the basic atom model when an electron becomes excited (i.e. absorbs a photon or collides with a nearby atom or particle) and moves into an energy state greater than its ground state, must it always eventually...

Homework Statement The Attempt at a SolutionSince A is a vector in V and since the A_i form a basis, we can write A as a linear combination of the A_i. We write A = x_1 A_1 + ... + x_n A_n. Thus, we have, <x_1 A_1 + ... + x_n A_n,A_i> = 0 = x_1 <A_1,A_i> + ... + x_n <A_n,A_i>. Because...

Hello Forum, a lattice is a set of points. We can place a basis at each set of points. The basis can be one atom or a group of atoms. I thought that a translation of the basis would produce the whole crystal... How is a basis different from the unit cell? Are they the same thing...

Background:(you might not be interested so you can skip if you want) I am trying to learn general relativity using the Book Gravitation by Misner, Thorn and Wheeler. The book for the most part seems easy for me to understand but once in a while words i neither heared nor can find the meaning of...

Hello everyone, I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise. Specifically, I'm having a hard time understanding the matrix elements of the density matrix...

Hi everyone: How is the following derived? Just for example: \Deltax\alphae\alpha=\Deltax\alpha(\delta/\deltax\alpha) does it not mean? e\alpha=\delta/\deltax\alpha But How?

Question: The needed proposition and two examples: This is as far as I have got: I need to reduce this (I think) so I can represent is as a matrix! Any idea on how to do this? Thanks

Why should we want to write a vector in Rn in other than standard basis? A normal application of linear transformations in most textbooks is converting a given vector in standard basis to another basis. This is sometimes a tedious task. Why carry out this task? Thanks for your replies in advance.

Homework Statement Prove T is a linear transformation and find bases for both N(T) and R(T). Homework Equations The Attempt at a Solution T:M2x3(F) \rightarrow M2x2(F) defined by: T(a11 a12 a13) (a21 a22 a23) (this is one matrix) = (2a11-a12 a13+2a12)...

I'm trying to prove that every nxn matrix can be written as a linear combination of matrices in GL(n,F). I know all matrices in GL(n,F) are invertible and hence have linearly independent columns and rows. I was thinking perhaps there is something about the joint bases for the n-dimensional...

Is it possible to express ANY observable A(X,P) in terms of the ladder operators? I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be...

The finite case is fine, as a vector space it is easy to show that R^X is isomorphic to R^n. What about when X is infinite? I believe it is true in general that dim(R^X) = #(X), which I hope holds in the infinite case too. I know that the set given by B={b_x; x in X} defined as b_x(y) =...

Homework Statement I have a problem on my assignment in which I am required to find the specific heat of a two atom basis (diatomic) using the Debye model. My problem is coming up with the density of states for a diatomic setup in 1D. Homework Equations Density of state...

Are there any theories with which the mass effect fields of Mass Effect and the slipstream space of Halo draw inspiration from? Is there any scientific basis for either, or is it complete fantasy?

I only possesses a rudimentary understanding of Linear Algebra so I'm not going to be rigorous in my explanation, but is the concept of an infinite basis well defined? More specifically, I was thinking about how the polynomials could form a basis for function space, given that every function has...

Hello everybody, I am given a "Sobolev type innerproduct" \langle f,g \rangle_{\alpha} = \langle f,g \rangle_{L^2} + \alpha \langle Rf,Rg \rangle_{L^2} for some \alpha \geq 0 and R some differential operator (e.g. the second-derivative operator). My question is now whether a function...

Hi all, Say that I have a 1D signal such that f=Bw where f is the signal B is the basis functions and w is the wave co-efficients. The question that I have is how do I find the B matrix in Matlab. I am looking through WaveLab and Rice Wavelet packages but simply cannot find an answer. As...

Hi everyone, This not a homework question. I'm reviewing some linear algebra and I found this on a worksheet. I just need a hint on how to approach this problem. Let β=\{ v_1,v_2,...,v_n\} be a basis for R^n . Let M be the matrix whose columns are the basis vectors in β. Do the...

I'm currently doing a self study course on Linear Algebra. Can anyone give me an example of vector space and basis with reference to Structural Engineering? For example I have a displacement vector for a simply supported beam as: [thata_a theta_b]^T where; theta_a and theta_b...

What is the basis for the theory that WIMPs could be detected by seeing a vibration in the atomic nucleus of normal matter? If they (all) really do interact so weakly, why do scientist think they might be able to detect just a few?? an explanation in layman's terms would be great. thanks?

I am working on a problem where I want to approximate a transcendental function of the form f(x) = x^Ne^{x} for x \geq 0 as a linear combination of functions of the form x^v \text{where} -1 < v < 0. How can I find the basis functions of the desired form to represent my transcendental...

In my Abstract Algebra course, it was said that if E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}. The basis of E over \mathbb{Z}_{3} is equal to [1,\bar{X}]. But this, honestly, doesn't really make sense to me. Why should \bar{X} be in the basis without it containing any other \bar{X}^n...

Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates...

Hi I'm stuck on this problem and I could not find similar examples anywhere.. any help would be greatly appreciated, thank you. Homework Statement Compute the change of basis matrix that takes the basis V1 = \begin{bmatrix} -1 \\ 3 \end{bmatrix} V2 = \begin{bmatrix} 2 \\ 5 \end{bmatrix}...