Basis Definition and 1000 Threads

Homework Statement find a basis of the kernel of the matrix that 1 2 0 3 5 0 0 1 4 6Homework Equations how the vectors are linearly independent and span the kernel The Attempt at a Solution Does it mean I need to samplify the 1 2 0 3 5 0 0 1...

Hi, I'm working through Schutz's intro to GR on my own, and I'm trying to do problems as I go to make sure it sinks in. I've encountered a bump in chapter 5, though. I don't think this is a tough problem at all, I think it's just throwing me off because x and y are coordinates as well as...

Hello. I really need help with this one: Homework Statement I have a 3 dimensional state space H and its subspace H1 which is spanned with |Psi> = a x1 + b x2 + c x3 and |Psi'> = d x1 + e x2 + f x3 Those two "rays" are linearly independent and x1, x2, and x3 is an...

Homework Statement lets say i have a matrix A which is symmetric i diagonalize it , to P-1AP = D Question 1) am i right to say that the principal axis of D are no longer cartesian as per matrix A, but rather, they are now the basis made up of the eigen vectors of A? , which are the columns...

Homework Statement Write the A matrix and the x vector into a basis in which A is diagonal. A=\begin{pmatrix} 0&-i&0&0&0 \\ i&0&0&0&0 \\ 0&0&3&0&0 \\ 0&0&0&1&-i \\ 0&0&0&i&-1 \end{pmatrix}. x=\begin{pmatrix} 1 \\ a \\ i \\ b \\ -1 \end{pmatrix}. Homework Equations A=P^(-1)A'P. The...

can anyone tell me on what basis do we assign ok not assign conclude that 1s orbital has a "less energy" than 2s..? what do we really mean by saying less energy

Homework Statement In a given basis \{ e_i \} of a vector space, a linear transformation and a given vector of this vector space are respectively determined by \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0\\ 0&0&5\\ \end{pmatrix} and \begin{pmatrix} 1 \\ 2 \\3 \end{pmatrix}. Find the matrix...

I'm moving on to my next section of work and i come across this example: Consider the homogeneous system x + 2y − z + u + 2v = 0 x + y + 2z − 3u + v = 0 It asks for a basis to be found for the solution space S of this system. And also what is the dimension of S. I know this might be...

Hi, We know that the Pauli matrices along with the identity form a basis of 2x2 matrices. Any 2x2 matrix can be expressed as a linear combination of these four matrices. I know of one proof where I take a_{0}\sigma_{0}+a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{3}\sigma_{3}=0 Here, \sigma_{0} is...

Why is an orthogonal basis important?

Homework Statement Right, I know how to do questions on Jordan Normal Form and find a basis, but there is one part I don't understand. Let's take for example this matrix, call it A. \begin{bmatrix} -3 & 1 & 0 \\ -1 & -1 & 0 \\ -1 & -2 & 1 \end{bmatrix} We find the characteristic...

Homework Statement A set of 6 vectors in R5 cannot be a basis for R5, true or false? The Attempt at a Solution I'm thinking true, because any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5. To be a basis it must be a linearly independent...

This is a very basic question in understanding General Relativity, but the answer still eludes me. The simples way I can state it is: "what exactly represent the 4 basis vector at a certain point on the manifold?" But let me explain myself. Let's take a 4 dimensional Minkowsky space. In this...

Hi everyone Homework Statement File at attachment. Given are two basis and the orthogonal matrix B. When r=...(see attachment) I shall proof that the lambdas are equal. Homework Equations - The Attempt at a Solution I have much trouble with this exercise and it is quite...

Homework Statement Let v=5t-2,S={v_1,v_2 }={t+1,t-1} is a basis of P_1 where P_1 is a vector space of all polynomials of degree ≤1. What is [v]s? Let v=5t-2,S={v_1,v_2 }={t+1,t-1} is a basis of P_1 where P_1 is a vector space of all polynomials of degree ≤1. What is [v]s? 2. The attempt at a...

curiosity about "complete" basis Hi In QM books , people talk about complete basis. I was checking some linear algebra books. Of course , we have a concept of basis in linear algebra. But these books nowhere talk about "complete" basis. Maybe math people have some more technical term for...

Hello all. This is my first post here. Hope someone can help. Thank you guys in advance. Here is the question: I have a n-by-n matrix A, whose eigenvalues are all real, distinct. And the matrix is positive semi-definite. It has linearly independent eigenvectors V_1...V_n. Now I have known...

Homework Statement Let V be the space spanned by f1 = sinx and f2 = cosx. (a) Show that g1 = 2sinx + cosx g2 = 3cosx form a basis for V. (b) Find the transition matrix from B' = {g1, g2} to B = {f1, f2}. Homework Equations P = [[u'1]B [u'2]B...[u'n]B] The Attempt at a Solution...

I have the following very basic question, and i'd really like your help! If we have a system, that is described by a Hamiltonian H, then we can expand the state of the system to the basis of H. And we say that, if we measure the observable H the state will collapse to one of H's eigenstates...

Homework Statement ... R4 consisting of all vectors of the form [a+b a c b+c] Homework Equations Gram-Schmidt process, perhaps? The Attempt at a Solution Not sure how to approach this one. Helpful hint?

I'm trying to create a circle in 3D based off of 4 inputs. Position1 Position2 LineLength1 LineLength2 The lines start at the positions, and they meet at their very ends. To do this I've gotten the distance between the points, found the radius of the circle, the position of the center of the...

I am trying to understand the notions of rank of an R-Module, free-module, basis, etc. I would like to understand this line (expand on it, find some critical examples/counterexamples ,etc) that I am quoting from Dummit & Foote: "If the ring R=F is a field, then any maximal set of...

I am a physicist, so my apologies if haven't framed the question in the proper mathematical sense. Matrices are used as group representations. Matrices act on vectors. So in physics we use matrices to transform vectors and also to denote the symmetries of the vector space. v_i = Sum M_ij...

Hello all, I'm currently working on a problem in which I'm attempting to characterize a centered Gaussian random process \xi(x) on a manifold M given a known covariance function C(x,x') for that process. My current approach is to find a series expansion $\xi(x) = \sum_{n=1}^{\infty} X_n...

There has been much discussion of time travel, but I haven't yet found an answer to a question I have. That is, what are the specific theories or factors that allow for the possibility of time travel? The issue has been discussed and debated many times on this forum and I'm not so much...

Let β={u1, u2, ... , un} be a subset of F^n containing n distinct vectors and let B be an nxn matrix in F having uj as column j. Prove that β is a basis for Fn if and only if det(B)≠0. For one direction of the proof I discussed this with a peer: Since β consists of n vectors, β is a...

Hi Everyone, I want to ask if I did this problem correctly. Homework Statement Find a orthogonal basis for subspace {[x y z]T|2x-y+z=0} Homework Equations X1= [3 2 -4]T, X2=[4 3 -5]T The Attempt at a Solution Gram-Schmidt: F1=X1= [3 2 -4] F2= X2- ((X2.F1)/||F1||2)F1= [4 3...

Homework Statement Find the basis for the row space The Attempt at a Solution the given matrix is 0 1 2 1 2 1 0 2 0 2 1 1 So i reduced to row-echeleon form 2 1 0 2 0 1 2 1 0 0 3 1 so then rank = 3. My textbook states that the basis of the row space are the row vectors of leading ones...

Homework Statement Let S = [x y z w] \in R^4 , 2x-y+2z+w=0 and 3x-z-w=0 Find a basis for S. Homework Equations The Attempt at a Solution I started by putting the system into reduced row form: [2 -1 2 1] [3 0 -1 -1] [2 -1 2 1] [0 3 -8 -5] [6 0 -2 -2]...

I have been thinking much on the nature of pure mathematics. I believe this forum would make the best place to post over say the philosophy section, as i am more interested in the opinions of working mathematicians and physicists than philosophers. In my opinion pure mathematics is the core...

Homework Statement It's not a homework question but a doubt I have. Say I want to write \vec A \times \vec B in the basis of the cylindrical coordinates. I already know that the cross product is a determinant involving \hat i, \hat j and \hat k. And that it's worth in my case...

In my book I see that the author finds the jordan basis for matrix A={(-3,9),(1,3)} by finding nullspace for (A-3I) and (A-3I)^2 without any justification, do I miss something trivial here?

Hello! I have been contemplating this question for a few days now and I am interested to know if anyone here has any input on the matter, and to critique my reasoning. There are other threads on this, none of which, from what I could find, made a distinction between homosexual behaviour and...

Consider a) f1=1, f2=sinx , f3=cosx b) f1=1, f2=ex , f3=e2x c)f1=e2x , f2=xe2x f3=x2e2x in each part B={f1,f2,f3} is a basis for a subspace V of the vector space. Find the matrix with respect to B of the differentiation operator D:V→V

i want to extend the set S={(1,1,0,0),(1,0,1,0)} to be a basis for R4. I know I am going to need 4 vectors, so i need to find 2 more that aren't linear combinations of the first 2. Is there a better way to approach this other than choose 2 at random and check linear independence/dependence...

does multiplying the invertible matrix A to the basis {X1,X2,X3..Xn} create a new basis; {AX1,AX2,Ax3..AXn}? where Xn are matrices I can prove that for eg if {v1,v2,v3} is a basis then {u1,u2,u3} is a basis where u1=v1 u2=v1+v2, u3=v1+v2+v3 I setup the equation c1(u1)+c2(u2)+c3(u3)=0 and...

With normal vectors i usually check there is the correct number of vectors i.e 3 for R3 2 for R2 etc and then just check for linear independence but reducing the matrix that results from c1v1+c2v2+..cnvn=0 and determining of unique solution or infinite solutions. There are the right number of...

How to prove that two reciprocal basis are either both right ended or both left-handed? If (e_1,e_2,e_3) and (e^1,e^2,e^3) are two such basis, since the scalar triple products depend on orientation, it would be enough to show that VV'=1 (where V and V' are the volumes, taken with their sign, of...

I see the term dictionary used a lot and it sounds a lot like a basis for a vector space. But what is the different? Can we collection of vectors be both a basis and a dictionary? Thanks

Homework Statement A particle with spin 1/2 and magnetic moment is in a magnetic field B=B_0(1,1,0). At time t=0 the particle has the spin 1/2 \hbar in the z direction. i) Write the hamiltonian with respect to the basis that is defined by the eigenvectors of \widehat{S}_z Homework...

Hi there, I have 23D subspace, defined by an equation (hyperplane) c1*x1 + ... + c24*x24 = 0; I wonder if there is an automated way to find basis of the subspace? I have access to Maple and Matlab. Thanks.

Water "divining" or "witching". What is the basis behind it? Before I start, let me say that I am hugely skeptical but we do have an issue where finding a good hitting water well might be tough so thought I might check...lol For those that are unaware, people are basically using pieces of...

Hey! If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form (g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1) what causes Christoffel symbols to vanish and puts Riemann...

Homework Statement I'm stuck on how to start this. The Hammin metric is define: http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view&current=hamming_metric.jpg and I'm asked to: http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360 a) prove...

I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it? for example, i know...

Hi again, I don't want it to seem like I'm spamming topics here, but I was hoping I could get help with this dillema, too. So, let's say that, in affine 2-dimensional space, we have some two, non-orthogonal, independent vectors, and we also pick some point for an origin O. This clearly...

i am still confused how to prove that a set is a basis other than proving it linearly independent and system of generator that have to do with matrices? please help

How do I calculate the Basis for Im(T)? I am having troubles finding an example that will best fir here. I know that the I=diagonal matrix with all of all of the i=j entries being 1. Beyond that I am rather confused and don't know where I need to start.

If you find the exact same basis for two vector spaces, then is it true that the vector spaces are equal to each other?

Homework Statement A matrix a is idempotent if a^2=a. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents 2. The attempt at a solution the vector space in question is dimension 4, so I need to find 4 idempotent matrices. but i don't want to find them...