Basis Definition and 1000 Threads

Hi guys Say we are looking at a two-particle system consisting of two electrons (fermions). In my book it says that the basis states are given by \left| {\psi _{\alpha ,i} (r_m )} \right\rangle \left| {\psi _{\beta ,j} (r_n )} \right\rangle where rm and rn denote the two particles...

Homework Statement Given x in the interval [0, \pi], let \phi_{0}(x) = 1, and \Phi_{n} (x) = sin ((2n-1)x). Show that there are constants: {A_{n}}^{n=0}_{\infty} and {B_{n}}^{n=0}_{\infty} such that: \sum^{n=0}_{\infty}A_{n}\phi_{n}=\sum^{n=0}_{\infty}B_{n}\phi_{n} But A_{n}...

suppose that vectors in R3 are denoted by 1*3 matrices, and define T:R4 to R3 by T9x,y,z,t)=(x-y+z+t,2x-2y+3z+4t,3x-3y+4z+5t).Find basis of kernel and range.

Homework Statement find the basis of a subspace of R^3 spanned by S: 1. S = { (4,4,8) (1,1,2) (1,1,1)} 2. S = { (1,2,2) (-1,0,0) (1,1,1) Homework Equations Im allowed to use calculator. The Attempt at a Solution Im not really sure what this is about. . .I tried the following...

Homework Statement Find the basis and dimension of the following subspace U of P2 p(x) \ni P2 such that p(1) = p(2)Homework Equations The Attempt at a Solution I know all quadratics are in the form ax2 + bx + c set p(2) = p(1) 4a + 2b + c = a + b + c b = -3a Therefore ax2 -3a + c Basis(U)...

Homework Statement To test my knowledge of Sakurai, I asked myself to: "Prove that an operator being unitary is independent of basis." The Attempt at a Solution I want to show the expansion coefficients’ squared magnitudes sum to unity at time “t”, given that they do at time t = t0...

Hey i was wondering how to express the time evolution operator U(t,to) to a momentum eigen state |p> for a particle moving in the xdirection under a zero potential, V= 0. The reason i need this is that iam told the only way to get the matrix element of the time evolution operator using position...

I'm curious to know why chemists like to use Gaussian basis set in case of an ab-initio (ex.DFT) calculation. I understand that the molecules that are of interest to chemists are non-periodic and hence plane wave basis is not useful, but can't they use other real space basis like a grid? What...

Why is it enough to prove that a set of vectors is a BASIS to a FINITE DIMENSIONAL Vector Space, it is enough to show that it is Linearly Independent. No Need to prove that it spans the whole vector space?

Homework Statement Find a basis and dimension to each of the following subspaces of R4: U = {(a+b,a+c,b+c,a+b+c)|a,b,c∈R} Homework Equations The Attempt at a Solution I started by making a linear system. w(a + b) + x(a + c) + y(b + c) + z(a + b + c) = 0 a(w + x + z) + b(w...

I got a quick question about the transformation matrix from the spin-z basis to the spin-x basis for spin-1/2 particles. Would the matrix be: \left(\begin{array}{ccc} \frac{e^{i\theta}}{\sqrt{2}} &\frac{e^{i\delta}}{\sqrt{2}} \\ \frac{e^{i\theta}}{\sqrt{2}} & -\frac{e^{i\delta}}{\sqrt{2}}...

Hello, let's consider, for example, the Clifford algebra CL(2,0) and the following mapping f for an arbitrary multivector: a + b\mathbf{e_1}+c\mathbf{e_2}+d\mathbf{e_{12}} \longmapsto a\mathbf{e_{12}} + b\mathbf{e_1}+c\mathbf{e_2}+d For vector spaces R^n we can permute the coordinates of...

Homework Statement I need to find a basis for the following: S = {f are polynomials of degree less than or equal to 4| f(0) = f(1) = 0} 2. The attempt at a solution A general polymial is of the form: p(x) = ax^4 + bx^3 + cx^2 + dx + e Now for p(0) = p(1) = 0 I must have: e = 0 and a + b...

Homework Statement Let P_4(\mathbb{R}) be the vector space of real polynomials of degree less than or equal to 4. Show that {{f \in P_4(\mathbb{R}):f(0)=f(1)=0}} defines a subspace of V, and find a basis for this subspace. The Attempt at a Solution Since P_4(\mathbb{R}) is...

Homework Statement The solutions to the linear differential equation d^2u/dt^2 = u for a vector space. Find two independent solutions, to give a basis for that solution space. The Attempt at a Solution I want to understand this question. I feel that there's something I'm missing. I...

Homework Statement Homework Equations The Attempt at a Solution I don't know how to do this problem.. Help please!

Homework Statement Given a linear transformation F: R^3 --> R, F(x,y,z) = 3x-2y+z, find I am (F) and dim (Im (F)) Homework Equations I have found that dim(ker F) = 2 and from the theorem dim (V) = dim (Ker F) + dim (Im F), I know dim (V) = 3, so dim (Im F) = 1. The Attempt at...

3x1 + x2 + x3 = 0 6x1 + 2x2 + 2x3 = 0 -9x1 - 3x2 - 3x3 = 0 I'm not sure how to approach this problem. I've rewritten these equations as a matrix [3 1 1] [6 2 2] [-9 -3 -3] Reduced Echelon from gave me this [3 1 1] [0 0 0] [0 0 0] Am I approaching this the wrong way...

Homework Statement Since it's kind of hard to type out, I'll try to post a screenshot: [PLAIN]http://img841.imageshack.us/img841/7357/questionq.jpg Homework Equations There's the definition of a basis, vector space, and all the axioms. The Attempt at a Solution I understand...

I am having trouble proving this statement. Please help as I am trying to study for my exam, which is tomorrow Prove that the standard topology on RxR is equivalent to the one generated by the basis consisting of open disks. Thanks :)

Homework Statement Let X be the vector space of polynomial of order less than or equal to M a) Show that the set B={1,x,...,x^M} is a basis vector b) Consider the mapping T from X to X defined as: f(x)= Tg(x) = d/dx g(x) i) Show T is linear ii) derive a matrix...

Homework Statement For I = [a,b], define: P3(I) = {v: v is a polynomial of degree ≤ 3 on I, i.e., v has the form v(x) = a3x3 + a2x2 + a1x + a0}. How to show v is uniquely determined by v(a), v'(a), v(b), v'(b). Homework Equations The Attempt at a Solution I'm not exactly sure...

I have a homework problem where I am to find y_2 for a 2nd ODE, with y_1=x. I'm familiar with the process of: let y_2 = ux y_2- = u'x u y_2'' = 2u' + u''x substituting these terms into the 2ODE, then letting u' = v. When integrating v and u' to solve for u, do I need to include...

Suppose I have a basis for a subspace V in \mathbb{R}^{4}: \mathbf{v_{1}}=[1, 3, 5, 7]^{T} \mathbf{v_{2}}=[2, 4, 6, 8]^{T} \mathbf{v_{3}}=[3, 3, 4, 4]^{T} V has dimension 3, but is in \mathbb{R}^{4}. How would one switch basis for this subspace, when you can't use an invertible...

Any suggestions on how to go about becoming a researcher in this area? I have very little formal background in math and physics and will be studying this on my spare time (which means when I'm not working on the cognitive stuff, though I would love to integrate something more rigorous into...

About 6 minutes ago I thought of this, and I want to check if it is true. Is the kilogram and the second a basis for all units in existence? That is, can all units be derived from these two? I can't think of any other units that are independent.

I want to know what orthonormal basis or transformation physically means. Can anyone please explain me with a practical example? I prefer examples as to where it is put to use practically rather than examples with just numbers..

I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...

The two sets of matrices: {G_1} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 0 \\ 0 & 0 & { - 1} \\ 0 & 1 & 0 \\ \end{array}} \right){\rm{ }}{G_2} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 1 \\ 0 & 0 & 0 \\ { - 1} & 0 & 0 \\ \end{array}} \right){\rm{...

If we could find the Hamel basis for any infinite dimensional vector space, what kind of consequences would this have?

Hi all, I'm interested in proving/demonstrating/understanding why the Dirac gamma matrices, plus the associated tensor and identity, form a complete basis for 4\times4 matrices. In my basic QFT course, the Dirac matrices were introduced via the Dirac equation, and we proved various...

A = 1 2 -1 3 3 5 2 0 0 1 2 1 -1 0 -2 7 Problem: Find a basis for the row space of A consisting of vectors that are row vector of A. My attempt: I transpose the matrix A and put it into reduced row echelon form. It turns out that there are leading ones in every column...

Find the basis and dimension of the following homogeneous system: A = |1 0 2| |x1| |2 1 3| |x2| = [0,0,0] |3 1 2| |x3| My attempt: Solving the coefficient matrix for RREF, I get the identify matrix. So, x1=x2=x3=0 and the only solution is a trivial one. Does that mean...

Find a basis for the given subspaces of R3 and R4. a) All vectors of the form (a, b, c) where a =0. My attempt: I know that I need to find vectors that are linearly independent and satisfy the given restrictions, so... (0, 1, 1) and (0, 0, 1) The vectors aren't scalar multiples of...

As I read one linear algebra book I have, I am told that "If a vector space V has a basis with 'n' vectors, then every basis in vector space V has 'n' vectors. So every basis in R3 has 3, every basis in R4 has 4, etc. However, I have a problem that says: Let S = { "five vectors" } be a...

Homework Statement Well the same as the subject.. (a,b,c) is a Basis of R^{3} does (a+b , b+c , c+a) Basis to R^{3} I have another question .. is (a-b , b-c , c-a) Basis to R^{3} This is know is not true because if I use e1, e2 , e3 I got a error line. Homework Equations start of...

Homework Statement Consider the vector V= [1 2 3 4]' in R4, find a basis of the subspace of R4 consisting of all vectors perpendicular to V. Homework Equations I mean, I'm just completely stumped by this one. I know that in R2, any V can be broken down to VParallel + VPerp, which...

Homework Statement He made some notes, but I'm still confused. The Attempt at a Solution

In consideration of a lone wire element of differential length dL carrying a current I: Two things concern me here: 1) Is a magnetic field generated by current I really limited to only spaces that exist orthogonally to the line between the two ends of the wire element? If not, what does...

Homework Statement Prove that if m < n and if y_1,...,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j] = 0 for j = 1,..., m Homework Equations The Attempt at a Solution My thinking is somehow that we...

it is quite peculiar i know you do not want to embed the manifold into a R^n Euclidean space but still it is too peculiar it is hard to develop some intuition

Homework Statement Hi everyone. I'm studying a problem and I need to prove that I have a basis. I tryed a proof and to achieve it I need to show that : if k divides a*b and also divides a2 +2*b2 Then k divides both a and b. Homework Equations I'm not sure what I'm asserting is...

I'm trying to get a metric in the frame of a boosted observer. The spacetime in question has coframe and frame basis vectors \begin{align*} \vec{\sigma}^0 = \frac{-1}{\sqrt{F}}dt\ \ \ \ & \vec{e}_0 = -\sqrt{F}\partial_t \\ \vec{\sigma}^1 = \sqrt{F}dz\ \ \ \ & \vec{e}_1 =...

Homework Statement Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of equations. x - 2y + z = 0 y - z + w = 0 x - y + w = 0 Homework Equations The Attempt at a Solution (a) [1 -2 1 0] => [1 0 -1 2] [0 1 -1 1] => [0 1...

Homework Statement Let A = 1 3 2 2 1 1 0 -2 0 1 1 2 Viewing A as a linear map from M_(4x1) to M_(3x1) find a basis for the kernal of A and verify directly that these basis vectors are indeed linearly independant. The Attempt at a...

Hi, Here's my problem, probably not that difficult in reality but I don't get how to approach it, and I've got an exam coming up soon... An atom with total angular momentum l=1 is prepared in an eigenstate of Lx, with an eigenvalue of \hbar. (Lx is the angular momentum operator for the...

Homework Statement Define an inner product on P2 by <f,g> = integral from 0 to 1 of f(x)g(x)dx. find an orthonormal basis of P2 with respect to this inner product. Homework Equations So this is a practice problem and it gives me the answer I just don't understand where it came from...

Homework Statement Let S={v1=[1,0,0,0],v2=[4,0,0,0],v3=[0,1,0,0],v4=[2,-1,0,0],v5=[0,0,1,0]} Let W=spanS. Find a basis for W. What is dim(W)? Homework Equations The Attempt at a Solution i know that a basis is composed of linearly independent sets. This particular problem's...

Homework Statement Let M1 = [1 1] and M2 = [-3 -2] ________[1 -1]_________[ 1 2] Consider the inner product <A,B> = trace(transpose(A)B) in the vector space R2x2 of 2x2 matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R2x2 spanned by the...

L: R^4 => R^3 is defined by L(x,y,z,w) = (x+y, z+w, x+z) A) Find a basis for ker L We can re write L(x,y,z,w) as x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0). I then reduced it to row echelon form We now have the equations X-W=0 , Y+W=0, Z+W=0. There are infinitely many...