Change of basis Definition and 99 Threads

Homework Statement Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}. The part that I am confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't...

I have a density matrix in one basis and need to change it to another. I know the eigenvectors and eigenvalues of the basis I want to change to. How do I do this? Any help really appreciated- thanks!

Hello, I am doing calculation on change of basis vector. But I am unable to understand why we do it. I mean to say what is the use of it and where in physics or maths it is used. Can anybody please explain it?

Homework Statement Shanker 1.7.1 3.)Show that the trace of an operator is unaffected by a unitary change of basis (Equivalently, show TrΩ=TrU^{\dagger}ΩUHomework Equations I can show that via Shanker's hint, but I however can't see how a unitary change of basis links to TrΩ=TrU^{\dagger}ΩU...

Homework Statement Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance. If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and T(v2)=3v2. and B =...

Homework Statement https://dl.dropbox.com/u/4788304/Screen%20shot%202012-07-08%20at%2002.53.44.JPG This is the solution of Problem A.15 in Griffiths' Quantum Mechanics. Tx is the rotation matrix about x-axis for theta degrees; while Ty is the rotation matrix about y-axis for theta degrees...

Hello, I was wondering if the pseudoinverse can be considered a change of basis? If an m x n matrix with m < n and rank m and you wish to solve the system Ax = b, the solution would hold an infinite number of solutions; hence you form the pseudoinverse by A^T(A*A^T)^-1 and solve for x to...

Homework Statement The Attempt at a Solution So first I thought to myself that the proper way of doing this problem was to construct each of the standard basis vectors as a linear combination of the basis given us. I have, T(1,0,0) = \frac{1}{2} T(1,0,1) + \frac{1}{2} T(1,0,-1) =...

Hi, I'm working on an example question with the following info: \alpha = {(3,0,1) , (3,1,1), (2,1,1)} \beta = {(1,1), (1,-1)} Are a set of bases. [T]\beta\alpha = \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1 \end{bmatrix} Now they go on to say: Let T: R3--> R3 be the transformation whose...

Why should we want to write a vector in Rn in other than standard basis? A normal application of linear transformations in most textbooks is converting a given vector in standard basis to another basis. This is sometimes a tedious task. Why carry out this task? Thanks for your replies in advance.

Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates...

Hi I'm stuck on this problem and I could not find similar examples anywhere.. any help would be greatly appreciated, thank you. Homework Statement Compute the change of basis matrix that takes the basis V1 = \begin{bmatrix} -1 \\ 3 \end{bmatrix} V2 = \begin{bmatrix} 2 \\ 5 \end{bmatrix}...

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 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 B1 = {[1,2], [2,1]} is a basis for R2 B2 = {[1,-1], [3,2]} is a basis for R2 Find the change of basis matrix from B1 to B2 Homework Equations [B2 | B1] The Attempt at a Solution For some reason I can not solve this. I keep ending up with the matrix...

hi.. can anyone say what is the concept behind change of basis.. y do we change a vector of one basis to another?

Homework Statement Could someone help me understand the following manipulations concerning change of babsis in an N-dimensional space: |i'\right\rangle=R|i\right\rangle=\sum_{j=1}^NR_{ji}|j\right\rangle multiply around by (R^{-1})_{ik}...

Making a change of basis in the matrix representation of a linear operator will not change the eigenvalues of that linear operator, but could making such a change of basis affect the geometric multiplicities of those eigenvalues? I'm thinking that the answer is "no", it cannot.. Since if...

(a) Let A (matrix) =c1= [1,2,1], c2 = [0,1,2], c3 = [3,-2,-1] be a matrix (c1,c2,c3 refer to the columns of the matrix A, which is a 3x3 matrix) expressed in the standard basis and let w1 = (0,0,1)T, w2 = (0,1,2)T , w3 =(3,0,2)T , find the vector AUE in w basis. (b). Referring to problem (a)...

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...

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

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 &...

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...

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...

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...

Homework Statement Find the matrix of f relative to Alpha' and Beta'. Alpha' = [(1,0,0), (1,1,0), (2,-1,1)] Beta' = [(-13,9), (10,-7)] The question originally reads that f is a bilinear form. I've found a (correct according to answer key) matrix A that is 3 -4 4 -5 -1 2...

Hi guys 1) We are looking at a Hamiltonian H. I make a rotation in Hilbert space by the transformation {\cal H} = \mathbf a^\dagger\mathsf H \mathbf a = \mathbf a^\dagger \mathsf U\mathsf U^\dagger\mathsf H \mathsf U\mathsf U^\dagger\mathbf a = \mathbf b^\dagger...

Homework Statement Suppose that for each v = (x1, x1, ... xn) in Rn, q(v) = XTAX for the given matrix A. For the given basis B of Rn, find the expression for q(v) in terms of the coordiantes yi of v relative to B. a) A =...

Linear Algebra - Change of basis matrices and RREF question what in the world?? Homework Statement Suppose the linear transformation T: P3 -> P2, over R has the matrix A = \begin{bmatrix}1&2&0&0\\0&1&2&1\\1&1&1&1 \end{bmatrix}...

Homework Statement Let A = E4 in R4 (standard basis) and B = {x^2, x, 1} in P2 over R. If T is the linear transformation that is represented by \begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix} relative to A and B, find...

Homework Statement Let A = {(1, 1), (2,0)} and B = {(0, 2), (2, 1)} in R2. a) Find [u]A (u with respect to A) if [u]B = [3, -2]. Homework Equations The Attempt at a Solution I tried to find [I]AB (transition matrix from B to A), then apply to [u]B, but couldn't represent (2, 1)...

Homework Statement Let {e1,e2,e3} be a basis for the vector space V, and T:V \rightarrow V a linear transformation. let f1 ;= e1 f2;=e1+e2 f3;=e1+e2+e3 Find the Matrix B of T with respect to {f1,f2,f3} given that the matrix with respect to {e1,e2,e3} is \[ \left(...

Homework Statement I need to prove this formula, but I'm not sure how to prove it.[T]C = P(C<-B).[T]B.P(C<-B)-1 whereby B and C are bases in finite dimensional vector space V, and T is a linear transformation. Your help is greatly appreciated! Homework Equations T(x)=Ax [x]C=P(C<-B)[x]B...

Homework Statement Let E={1, x, x2,x3} be the standard ordered basis for the space P3. Show that G={1+x,1-x,1-x2,1-x3} is also a basis for P3, and write the change of basis matrix S from G to E. Homework EquationsThe Attempt at a Solution Here's what I got: S_E^G=\left( \begin{array}{cccc}...

I would like to know how, given a metric with non-zero off-diagonal components g_{mn}, m \neq n, one can find if another (orthonormal or null) frame exists in which g_{\mu \nu}=constant for all components of the metric. And if it exists how to compute this basis. Thanks!

change of basis [x]B + [y]B = [x+y]B[ Homework Statement Let B = {v1,...vn} be a basis for a vector space V, and let x = a1v1 + ... anvn and y = b1v1 + ...+ bnvn be arbitrary vectors in V. Find [x]B, [y]B and [x+y]B Homework Equations [x]B + [y]B = [x+y][SUB]B The Attempt at a...

So...I've got an operator. Omega = (i*h-bar)/sqrt(2)[ |2><1| + |3><2| - |1><2| - |2><3| ] Part a asks if this is Hermitian, and my answer, unless I'm missing something, is no. Because the second part in square brackets is |1><2| + |2><3| - |2><1| - |2><3| which is not the same as Omega...

Homework Statement Let B & C be the following subsets of R^2 B= {[3 1] , [2 2]} (the vectors should be in columns instead of rows) C= {[1 0] , [5 4]} Let T: R^2 -> R^2 be the linear transformation whose matrix with respect to the basis B is [2 1] [1 5] (the brackets should be joint...

Homework Statement Let e_{i} with i=1,2 be an orthonormal basis in two-dimensional Euclidean space ie. the metric is g_{ij} = \delta _{ij}. In the this basis the vector v has contravariant components v^{i} = (1,2). Consider the new basis e_{1}^{'} = 5e_{1} - 2e_{2} e_{2}^{'} = 3e_{1} - e_{2}...

Homework Statement In \mathcal{P}_3 with basis B = \langle 1 + x, 1 - x, x^2 + x^3, x^2 - x^3 \rangle we have this representation. \text{Rep}_B(1 - x + 3x^2 - x^3) = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 2 \end{pmatrix}_B Find a basis D giving this different representation for the same...

I started working on this topic and thought I would post it for other to contribute their ideas. The Clebsch-Gordan coefficients basically tell you how to move between two different representations of spin. I am trying to work them out logically. For starters I am doing the very simplest case...

Hello! I was wondering if someone can tell me about any application to change of basis... The application can be of any sort, though. Thanks!

consider the basis B={1,x,x^2} and B'={1,1-x,x^2-4x+2} for R2[x]. Find the change of basis matricses [id]B'toB and [id]BtoB' Really stuck on this! anyone can help me please?

find the change of basis matrix PC<->B from the given ordered basis B to the given ordered basis C of the vector space V: V=R^2; B={(-5,-3),(4,28)}; C={(6,2),(1,-1)} The Attempt at a Solution I'm having a hard time grasping the concept of changing basis. Can someone please...

Will a set of vectors stay linearly independent after a change of basis? If it's not always true then is it likely or would you need a really contrived situation?

Homework Statement Consider the 2-dimensional complex vector space V of functions spanned by sin x and cos x. For a fixed real number α, define a linear operator T ≡ Tα on V by putting T(f(x)) = f(x + α). Find the matrices [T]B and [T]E of T relative to the bases B = {cos x, sin x} and E =...

Actually after I wrote down the query on the invertible matrix which I posted a few days ago I happened to refer again to Kunze Huffman and found that this is a standard theorem regarding transformation of linear operator from one basis to another. Then I realized that the point which was...

Hi there, just doing some basic linear algebra for quantum computation / quantum information theory, and am wondering whether I'm changing the basis of an operator correctly. If I have two orthogonal basis vectors of space C2 given by (~ = complex conjugate): S1 = [|0>, |1>] and S2 =...

Hello all, I need some help... If I know the form of a wavefunction in the S_z basis, say it is spin up, how do I convert that to a wavefunction expressed in the S_x basis? Is there a very simple way to do this? Thanks