(5) Let A, B be two 3 x 3 matrices with eigenvectors v1, 2, vs and...
Problem 4. Let B = {V1, 02, 03} CR, where [3] [1] 01 = 12, 02 = 12103 = 1 [1] [2] 4.1. Show that the matrix A = (v1 V2 V3) E M3(R) is invertible by finding its inverse. Conclude that B is a basis for R3. 4.2. Find the matrices associated to the coordinate linear transformation T:R3 R3, T(x) = (2]B- and its inverse T-1: R3 R3. Use your answers to find formulas for the vectors 211 for...
5. Given a linear map f R3R3 if V Vi, V2, va) is a basis of R3, and further, a) State the defining matrix of f under the basis vi, V2, vs) -3 2 0 b) Let W-(w1, w2, w3) be another basis of R3 and P42 be the change- 01-1 of-coordinate matrix from V to W. Let A be the defining matrix for f under the basis W diagonalize A.
5. Given a linear map f R3R3 if V...
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic field. σ2 10), (a) Find the eigenvalues and corresponding eigenvectors for all three Pauli spin matrices. Show all of vour work (b) In Linear Algebra, two matrices A and B are said to commute if AB BA and their commutator defined as [A,...
Let A be a 2x2 matrix with eigenvalues 5 and 3 and corresponding eigenvectors V1 = | Let {XK) be a solution of the difference equation asmenn :)--[;)] wywood 11 **+1 = Axx, Xo = a. Computex, = Axo. (Hint: You do not need to know A itself.] b. Find a formula for xk involving k and the eigenvectors V, and V2.
2. 15 points Consider the following matrices 2 -2 2 3 -3 3 4 -2 2 4 6 2 12 2 2 36 1 1 18 3 3 53 -3 1 -22 2 21 33 -1 1 -1 1 ,ws 3 3 -2 3 -1 0 -20 0 -1 3 7 -3 2 Let V span^v1, v2, v3) and W-span{w1, w2, w3, wa,w5, ws). (a) By finding more suitable bases, give a simple description of the subspaces V and W....
-247 -3 2. Let V1 = 1 , V , and V3 = , let B = (V1, V2, V3), and let W be the subspace spanned -2 by B. Note that B is an orthogonal set. 21 with respect to B, without inverting any matrices or a. [1 point] Find the coordinates of ū= 1: L 6 solving any systems of linear equations. 5 637 10 16. it point Find the sector in We st o b. [1 point]...
[5] (c) Let A and B be two 3x3 matrices, and let X = Suppose further that the linear system BX = 2 has infinitely many solutions. How many solutions does the linear system have? Justify your answer! (Hint: use det(B) and det(AB).]
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
Let A, B, C, and D be matrices with the following sizes: A, 5×3 B, 3×2 C, 3×5 D, 1×3 Which of the following matrix operations are defined? i) AB (ii) A + 1 4 C (iii) DC