Basis Definition and 1000 Threads

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 Let V=(x1,x2,..Xn) Sum X = 0 be a subspace of R^n. Find a basis of V such that Sum X^2=1 Homework Equations The Attempt at a Solution for Sum X =0 x1+x2+..+xn =0 xn= -x1-x2-... So <x1, x2, ...,-x1-x2-...>=<x1, 0,0...,-x1> +<0,x2,0,...,-x2)+...

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...

Homework Statement Find the basis for the subspaces of R3 and R4 below. Homework Equations A) All vectors of the form (a,b,c), where a=0 B) All vectors of the form (a+c, a-b, b+c, -a+b) C) All vectors of the form (a,b,c), where a-b+5c=0 The Attempt at a Solution I honestly had...

Homework Statement Find a basis for the solution space of the homogeneous systems of linear equations AX=0 Homework Equations Let A=1 2 3 4 5 6 6 6 5 4 3 3 1 2 3 4 5 6 and X= x y z...

Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...

Homework Statement Consider a linear transformation L from Rm to Rn. Show that there is an orthonormal basis {v1,...,vm} of Rm to Rn such that the vectors {L(v1),...,L(vm)} are orthogonal. Note that some of the vectors L(vi) may be zero. HINT: Consider an orthonormal basis {v1,...,vm} for...

hi guys, i have no idea of how to do the following question, could u give some ideas? Q:determine whether or not the given set forms a basis for the indicated subspace {(1,-1,0),(0,1,-1)}for the subspace of R^3 consisting of all (x,y,z) such that x+y+z=0 how should i start? i know the...

Homework Statement Given one known orthonormal basis S in terms of the standard basis U, how would I express a third basis T in terms of U when I know its representation in S? For example, U consists of <1,0,0> <0,1,0> <0,0,1> And S consists of (for example) <0.36, 0.48, -0.8>...

Hi, I understand that the maximum packing fraction for a particular atomic structure can be calculated assuming the nearset neighbours are touching but my question is how can the maximum packing fraction be calculated for a basis containing two different types of atoms? Thanks, James

Let S be the subspace of R^3 spanned by x=(1,-1,1)^T. Find a basis for the orthogonal complement of S. I don't even know where to start... I would appreciate your help!

Hi, I'm reading Shankar's Principles of QM and I find it not very clear on how exactly should I change basis of operator. I know how to change basis of a vector so can I treat the columns of operator matrix as vectors and change them? Or is it something else?

Homework Statement The set B = {-4-x^2, -8+4x-2x^2, -14+12x-4x^2} is a basis for P2. Find the coordinates of p(x) = (-2 +0x -x^2) relative to this basis. Homework Equations n/a The Attempt at a Solution so the set would be in a matrix like this: |-4 0 -1|...

I am having no luck understanding how to find the basis of a field adjoined with an element. For example Q(sqrt(2)+sqrt(3)) I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero. So, from that I know the...

We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3 a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai Thus a vector x in W_perp is x = (a,b,-ai) So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 +...

Homework Statement Recall that the matrix for T: R^{2} \rightarrow R^{2} defined by rotation through an angle \theta with respect to the standard basis for R^{2} is \[A =\begin{array}{cc}cos \theta & -sin \theta \\sin \theta & cos\theta \\\end{array}\]\right] a) What is the matrix of T...

Hi. Thanks for the help. Homework Statement Find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0. Homework Equations P' is the first derivative, P'' is the second derivative. The Attempt at a Solution The general form of a polynomial in P3 is ax^3+bx^2+cx+d...

Homework Statement My notes has the following statement, but I seem to have forgotten to write down the conclusion of the statement before my professor erased it from the board. "Any vector space V there will be a basis except for 1 type of space: " Any ideas as to what that 1 type of...

Homework Statement Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0. Homework Equations N/A The Attempt at a Solution Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be...