Basis Definition and 1000 Threads

Hello everyone, I am having difficulty understanding the difference between the basis of a subspace A and and the basis of the range of A. My textbook seems to follow the same approach in determining both. So are they essentially the same?

Homework Statement Find a basis for the solution space of the given homogeneous system. x1 x2 x3 x4 1 2 -1 3 | 0 2 2 -1 6 | 0 1 0 0 3 | 0 The Attempt at a Solution When I reduced to reduced row echelon form i get the following matrix...

Homework Statement Find a basis of U, the subspace of P3 U = {p(x) in P3 | p(7) = 0, p(5) = 0}Homework Equations The Attempt at a Solution ax3+bx2+cx+d p(7)=343a+49b+7c+d=0 p(5)=125a+25b+5c+d=0 d=-343a-49b-7c d=-125a-25b-5c ax3+bx2+cx+{(d+d)/2} -->{(d+d)/2}=2d/2=d...

Problem Given a transformation T : P(t) -> (2t + 1)P(t) where P(t) ϵ P3 (a) Show that transformation is linear. (b) Find the image of P(t) = 2 t^2 - 3 t^3 (c) Find the matrix of T relative to the standard basis ε = {1, t, t^2, t^3} (d) Find the matrix of T relative to the basis β1 = {1...

Homework Statement Find the basis of the solution space W \subset \Re^{4} of the system of linear equations 2x_{1} + 1x_{2} + 2x_{3} +3x_{4} =0 _{ } 1x_{1} + 1x_{2} + 3x_{3} = 0 Homework Equations The basis must span W and be independent. The Attempt at a Solution Solving...

Homework Statement Show that if { v_1, ... , v_k} spans V then {T(v_1), ... , T(v_k)} spans T(v) Homework Equations The Attempt at a Solution So we know that every vector in V can be written as a linear combination of v_1,...v_k thus we only need to show that {T(v_1)...

Homework Statement Given the matrix 0 1 0 0 0 1 -3 -7 -5 Find the eigenspaces for the various eigenvalues Prove that there cannot be a basis of R3 consisting entirely of eigenvectors of AHomework Equations The Attempt at a Solution The...

Homework Statement http://en.wikibooks.org/wiki/Linear_Algebra/Vector_Spaces_and_Linear_Systems/Solutions Problem 14 Can answer be (3,1,2)T (2,0,2)T? also, can I reduce the matrix without transpose? thanks Homework Equations The Attempt at a Solution

Homework Statement So the question is a map T: R^2x2 ---> R^2x2 by T(A) = BAB, where B = (1 1) (1 1) so i made A = (a c) and T(A) = ((a+b) + (c+d) (a+b) + (c+d))...

Homework Statement This is from my first-quarter graduate QM course. Part 4 of this problem asks me to compute the unitary operator U which transforms Sn into Sz, where Sn is the spin operator for spin 1/2 quantized along some arbitrary axis n = icos\phisinθ + jsin\phisinθ + zcosθ.Homework...

Homework Statement The matrix A = 1 1 1 1 -1 0 1 0 1 2 3 2 Express null space and row space of A in terms of their basis vectors. 2. The attempt at a solution I have found the null space to be: x3 [1 -2 1 0]^T + x4 [0 -1 0 1]^T. But my problem is how do i write the final answer correctly...

Homework Statement a) If U and W are subspaces of R^3, show that it is possible to find a basis B for R^3 such that one subset of B is a basis for U and another subset of B (possibly overlapping) is a basis for W. b) If U and W are subspaces of a finite-dimensional vector space V, show...

Homework Statement Find a basis for (1, a, a^2) (1, b, b^2) (1, c, c^2) Homework Equations The Attempt at a Solution M(1, a, a^2) + N(1, b, b^2) + K(1, c, c^2) = (0, 0, 0) M + N + K = 0 Ma + Nb + Kc = 0 Ma^2 + Nb^2 + Kc^2 = 0 This is as far as I got. I tried monkeying around with these 3...

I am unable to understand as to how the basis for the tangent space is \frac{\partial}{\partial x_{i}}. Can this be proved ,atleast intuitively? Bachman's Forms book says that if co-ordinates of a point "p" in plane P are (x,y), then \frac{d(x+t,y)}{dt}=\left\langle 1,0\right\rangle...

Homework Statement Are the following statements true or false? Explain your answers carefully, giving all necessary working. (1) p_{1}(t) = 3 + t^{2} and p_{2}(t) = -1 +5t +7t^{2} form a basis for P_{2} (2) p_{1}(t) = 1 + 2t + t^{2}, p_{2}(t) = -1 + t^{2} and p_{3}(t) = 7 + 5t -6t^{2}...

Homework Statement "In each of the given cases, decide whether the specified elements of the given vector space V (i) are linearly independent, (ii) span V, and (iii) form a basis. Show all reasoning. V is the space of all infinite sequences (a0, a1, a2, ...) of real numbers v1 =...

Homework Statement I am having trouble finding a basis in a given vector space. I understand how to find a basis of Rn, just find linearly independent vectors that span Rn But how would i find a basis of the set of 3x3 symmetric real matrices? Or Find a basis of real polynomials of...

Homework Statement Prove that symmetric and antisymmetric matrices remain symmetric and antisymmetric, respectively, under any orthogonal coordinate transformation (orthogonal change of basis): Directly using the definitions of symmetric and antisymmetric matrices and using the orthogonal...

Homework Statement [PLAIN]http://img193.imageshack.us/img193/3662/unledmcg.png The Attempt at a Solution I rewrote the whole thing in dictionary x_3 = 15 - 8x_1 - 4x_2 x_4 = 7 - 2x_1 - 6x_2 z = 0 + 22x_1 - 12x_2 x_i \geq 0 1\leq i \leq 4 a) So my basis/bases is x...

Homework Statement Is a set of orthogonal basis vectors for a subspace unique? The attempt at a solution I don't know what this means. Can someone please explain? I managed to find the orthogonal basis vectors and afterwards determining the orthonormal basis vectors, but I'm not sure what the...

i have the first solution y_1(t) = t for (1-t)y'' + ty' - y = 0. I need to get the 2nd linearly independent using Abels theorem. the integration is messy but i have it set up (sorry no latex); y_2 = (t) * integral to t ( 1/s^2 * exp( -integral to t (s(s+1) ds) ) ds. Could anyone...

Homework Statement (In textbook, given a figure, I cannot redraw that figure in this applet, so I shall describe the question in words) I am given a rectangular xy coordinate system determined by the unit basis vectors i and j and an x'y'-coordinate system determined by unit basis...

Homework Statement The four functions v0 = 1; v1 = t; v2 = t^2; v3 = t^3 form a basis for the vector space of polynomials of degree 3. Apply the Gram-Schmidt procedure to find an orthonormal basis with respect to the inner product: < f ; g >= (1/2)\int 1-1 f(t)g(t) dtHomework Equations ui =...

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...

Hi, All: Let Sg be the genus-g orientable surface (connected sum of g tori), and consider a symplectic basis B= {x1,y1,x2,y2,..,x2g,y2g} for H_1(Sg,Z), i.e., a basis such that I(xi,yj)=1 if i=j, and 0 otherwise, where I( , ) is the algebraic intersection of (xi,yj), e.g...

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...