Basis Definition and 1000 Threads

Many physicists claim that decoherence determines the emergence of the worlds in the Many World Interpretation (MWI). I have always found such a claim elusively proved and actually wrong. Recently I wrote a paper: http://arxiv.org/abs/1008.3708 addressing such a subject, and I sent it to...

Find a basis for the subspace S of R^3 consisting of all vectors of the form (a, 2a-b, b)^T, where a and b are real numbers Relevant equations would really just be the determinant of the system. I have tried so many 3x3 matrix combinations of the given form but no matter what the determinant...

1. In each case, find a basis of the subspace U: (a) U=span{[1 -1 2 5 1].[3 1 4 2 7],[1 1 0 0 0],[5 1 6 7 8]} (b) U=span{[1 5 -6]^T, [2 6 -8]^T, [3 7 -10]^T, [4 8 12]^T} 2. Determine if the following sets of vectors are a basis of the indicated space: {[1 0 -2 5]^T,[4 4 -3 2]^T,[0 1...

Homework Statement a) What is the mass of one mole of Ca(NO3)2? b) How many Ca2+ ions are there in 0.05 moles of Ca(NO3)2? c) How many moles ofNaCl are there in 450 g of this substance? (Avogadro’s number is 6.022*1023 1/mol.) Homework Equations The Attempt at a Solution a)...

Homework Statement We are given the following subspaces U := {x E R3: x1 + 2*x2 - x3 = 0} and V := {x E R3: x1 - 2*x2 - 2*x3 = 0} And we need to find a basis for (i) U (ii) V (iii) U n V (not an "n" but a symbol that looks like an upside-down U) (iv) span(U u V) (not a "u" but a symbol that...

Homework Statement Let B1 = {v1; v2; v3} be a basis of a vector space V and let B2 = {w1;w2;w3} where w1 = v2 + v3 ; w2 = v1 + v3 ; w3 = v1 + v2 Verify that B2 is also a basis of V and find the change of basis matrices from B1 to B2 and from B2 to B1. *Use the appropriate change of basis matrix...

Let V be the space of polynomials of degree 3 or less over \Re. For every \lambda\in\Re the evaluation at \lambda is the map ev_{\lambda} such that V \rightarrow \Re is linear. How do we find the coefficients of ev_{2} in the basis dual to \{1,x,x^2,x^3\}?

Hi guys Ok, so one way to define a Bravais lattice is to say that each lattice point can be reached by R = la1+ma2+na3 for some integer m, l and n. Obviously, this cannot be the case when we have a lattice with a basis. But does that also mean that a lattice with a basis does not have...

So, the rule for finding the matrix elements of an operator is: \langle b_i|O|b_j\rangle Where the "b's" are vector of the basis set. Does this rule work if the basis is not orthonormal? Because I was checking this with regular linear algebra (in R3) (finding matrix elements of linear...

Homework Statement Find a topological space which does not have a countable basis. Homework Equations Definition of basis : A collection of subsets which satisfy: (i) union of every set equals the whole set (ii) any element from an intersection of two subsets is contained in another...

Homework Statement So, I'm going through a proof and it is shamelessly asserted that the collection of clopen sets of {0,1}^{\mathbb{N}} is a countable basis. Can anyone reasure me of this, point me in the direction of proving it. Thanks Tal

Find a basis Beta in R^2 such that the beta matrix B of the given linear transformation T is diagonal. The Reflection T about the line R^2 spanned by [1 2], [1 2] is suppose to be verticle. B=S^-1AS or B=[[T(v1)]beta [T(v20]beta] so i found the reflection matrix to be...

Homework Statement The definition for local path connectedness is the following: let x be in X. Then for each open subset U of X such that x is in U, there exists an open V contained in U such that x is in V and the map induced by inclusion from the path components of V to the path components...

How can I show that trace is Invariant under the change of basis?

Homework Statement Let W be a 3x3 matrix where A^t(transpose)=-A. Find a basis for W. Homework Equations Find a basis for W. The Attempt at a Solution I have no idea how to start it.

The wikipedia article on connection forms refers to a local frame. What is the relationship between local frames and coordinate bases? Are they the same thing? Is one a subset of the other? The connection form article uses general notation e_\alpha for the basis elements instead of the...

Groebner basis calculations are an important technique for solving systems of nonlinear equations. Despite being (in the worst case), computationally intractable, they seem to be effective solutions for a multitude of problems. I have a problem that I am trying to solve. I have perused the...

Consider the collection of sets C = {[a,b), | a<b, and and b are rational } a.) Show that C is a basis for a topology on R. b.) prove that the topology generated by C is not the standard topology on R.So, I know for C to be a basis, there must be some x \in R, and in the union of some C1...

Homework Statement We are given 2 bases for V = \Re_{1 x 3}. They are \beta_{1} = \begin{bmatrix} 2 & 3 & 2\end{bmatrix} \beta_{2} = \begin{bmatrix} 7 & 10 & 6\end{bmatrix} \beta_{3} = \begin{bmatrix} 6 & 10 & 7\end{bmatrix} and, \delta_{1} = \begin{bmatrix} 1 & 1 &...

For example, say you start out with (2,1,0) and (2,0,2). Well the easiest answer here is to think of these two vectors in a plane, so you should take the cross product to get the vector that is not in the plane, and there you have a basis for R^3. But how about when we run into similar problems...

Hi, I was working through this proof in my linear al textbook and there's this one step I can't get past. Any help would be appreciated. Homework Statement Let V be a finite dimensional vector space, and let T be a linear map defined on V. ker T \subseteq V and I am T \cong V/kerT Let...

First of all I would like to wish a happy new year to all of you, who have helped us understand college math and physics. I really appreciate it. Homework Statement Determine the dimension of the image of a linear transformations f^{\circ n}, where n\in\mathbb{N} and...

Homework Statement The formulation of the problem confused me a little, so just to check. No T1 space has a locally finite space unless it is discrete. The Attempt at a Solution This means that, if X is a discrete T1 space, it has a locally finite basis, right? Btw, for the...

I was reading about dual spaces and dual bases in the book Linear Algebra by Friedberg, Spence and Insel (FSI) and they give an example of a linear functional, f_i (x) = a_i where [x]_β = [a_1 a_2 ... a_n] denotes the matrix representation of x in terms of the basis β = {x_1, x_2, ..., x_n} of...

A "countable basis" vs. "countably locally finite" problem Homework Statement Sometimes it's fairly difficult to name a thread for a specific problem. :smile: So, one needs to show that, if X has a countable basis, a collection A of subsets of X is countably locally finite of and only if...

Representing a vector in terms of eigenkets in continuos basis(stuck here,guys help) I was reading Dirac and there is this formula which bothers me, |P>= \int{|\right{\xi'd}\rangle{d\xi'}} + \sum{|\right{\xi^{r}b}\rangle} Where |\right\xi'\rangle denotes the eigenket corresponding to the...

Hello, 1. Proof: Let denote the number of representations of to the base . We must show that always equals 1. (this means that we are trying to prove that there is only one representation?) Line 2. Suppose that n = a_{0}k^{s} + a_{1}k^{s-1} + ... + a_{s-t}k^{t} then...

Hi, I'm scratching my head over the statement from my textbook which states when determinant is non-zero, the set of vectors blah blah is a basis for r^3. That does not make any sense to me because I know when a row of zeros in a matrix occur; the determinant is zero (through Gaussian...

I just want to test/verify my knowledge of change of basis in a linear operator.. (it's not a homework question). Suppose I have linear operator mapping R^2 into R^2, and expressed in the canonical basis (1,0), (0,1). Suppose (for the sake of discussion) that the linear operator is given by...

Homework Statement For this whole question let T be a linear transformation from R^3 to R^3 with T(1,0,0) = (2,2,2), T(0,1,0) = (1,2,2), T(0,0,1) = (0,0,1). (a) Find the image of (1,1,2009) (b) Find the matrix of T with respect to the standard basis in R^3 Homework Equations Standard...

Homework Statement Find a basis for the kernel space of the following matrix: -1 -2 -1 2 2 -2 -4 -4 10 2 1 2 2 -5 2 -1 -2 0 -1 0 row reduce to 1 2 0 1 0 0 0 1 -3 0 0 0 0 0 1 0 0 0 0 0 Somehow read the solution as { [-2 1 0 0 0]T, [-1 0 3 1 0]T } .. I don't...

Homework Statement Let T: R4 --> R7 be a linear map whose kernel has the basis of v = [1 0 1 2]T. What is the dimension of the image of T? The Attempt at a Solution I have a very loose understanding of kernel and image, and am trying very hard to get this question. From my understanding, the...

Homework Statement Let V be the vector space of all 2x2 matrices over Q V= {[x1 x2] : xi \in Q} ... x3 x4 Let A = [ -1 0 ] and let C:V --> V be the linear map C(X) = XA + AX .... -1 1 Find a basis for Ker(C) and a basis for Im(C) The Attempt at a Solution I used C(X) =...

Homework Statement Hi, i am trying to do the question on the image, Can some one help me out with the steps. [PLAIN]http://img121.imageshack.us/img121/6818/algebra0.jpg Solution in the image is right but my answer is so off from the current one. Homework Equations The...

I am a bit puzzled by the following. You know how they teach you that in order to find column space you just need to row reduce the matrix, look at the columns with leading 1's in them and then just read off those columns from the original matrix? Well, why does that actually work? I'm trying to...

Let A=2 -1 2 1 1 0 3 1 -2 1 0 1 -1 0 0 3 the characteristic polynomial of A is (x-1)3(x-2) find the minimal polynomial, jordan basis, and jordan normal form I know the minimum polynomial is (x-1)(x-2), but I am not sure how to find the nordan basis and jordan normal form

Hi! Homework Statement I can't for the life of me figure out how to do this. I need to find the basis for the span of these four vectors: V1= 3, 1, -2, -4 V2 = -5, -3, 5, 9 V3 = 5, -1, 0, -2 V4 = -1, 5 -6 -8 2. The attempt at a solution I've figured out that the determinant is...

Homework Statement [PLAIN]http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs574.ash2/149609_293915114994_507054994_1176494_3477051_n.jpg Homework Equations The Attempt at a Solution Ok so I know that this plane goes thru the origim I guess to find the two column vectors that span...

Homework Statement Find the eigen values, eigenspaces of the following matrix and also determine a basis for each eigen space for A = [1, 2; 3, 4]Homework Equations \det(\mathbf{A} - \lambda\mathbf{I}) = 0 The Attempt at a Solution OK, so I found the eigenvalues and eigenspaces just fine...

For a continuous eigen-basis the basis vectors are not normalizable to unity length. They can be normalized only upto a delta function. At the same time for discrete eigen basis the basis vectors are normalizable to unity length. What about the systems with both discrete as well as continuous...

Help! Find a basis. Find a basis for the subspace of R^4 spanned by, S={(6,-3,6,340, (3,-2,3,19), (8,3,-9,6), (-2,0,6,-5) Figured I would set up the linear combination to test for independence.

Find a basis for the subspace of R^4 spanned by, S={(6,-3,6,340, (3,-2,3,19), (8,3,-9,6), (-2,0,6,-5) Not too sure where to start.

I need some direction with respect to this problem please: Define the inner product on P4[x] over \Re as follows <f,g> = \int_{0}^{1}\f(x)g(x) dx let W be the subspace of P4[x] consisting of the poly. ) and all polynomials with degree 0, that is W =R Find a basis for...

Homework Statement Find a basis B of R^n such that the B matrix B of the given linear transformation T is diagonal. Reflection T about the plane x_1 - 2x_2 + 2x_3 = 0 in R^3. The Attempt at a Solution I just don't even know where to begin. I don't know how to interpret problem or how to...

Got another linear space question. I'm getting closer to understanding what's going on, but I'm not there yet. Homework Statement Find a basis for the space and determine its dimension. The space of all polynomials f(t) in P2 such that f(1) = 0. Homework Equations The Attempt at...

Homework Statement You're given two matrices (A and B). You want to find a basis for the space {x|x = Ay where By =0}. Homework Equations The Attempt at a Solution You're looking for all vectors x=Ay such that y is in the null space of B. So you're looking for a basis for only a part of...

Homework Statement Find the basis for the subspace S of the vector space V. Specify the dimension of S. S={a a+d} where a,d are elements of R and V= M2x2 {a+d d } Homework Equations I guess I know the standard basis for M2x2 are the [(10 00) (01 00) (00 10) (00 01)]...

Homework Statement So, if X has a countable basis {Bn}, then every basis C for X contains a countable basis for X. The Attempt at a Solution First of all, consider all the intersections of elements of C of the form Ci\capCj. For every x in the intersection (if it's non empty), choose a...

Hi, I have a question about linear transformation. So given a matrix A in the basis u (denoted as A_u). Now in another basis that I don't know, A_u becomes A_v. How can I find v? (I know u, A_u and A_v). Thank you very much,

Homework Statement Hi Say I have the kinetic energy operator denoted by T(ri) for the particle i. I wish to represent it in some \left| \sigma \right\rangle -representation. My book says it is given by T = \sum\limits_{\sigma _a ,\sigma _b } {T_{\sigma _a ,\sigma _b } \left| {\psi...