4. Consider the matrix A =10102 -1-1-12-1 01(a) [(b) [(c)Find the matrix of the orthogonal projection onto ker(A). Use this matrix tofind projker (A) (e1 e2 + e3).]Find an orthonormal basis for im(A). Find the distance from e3 to im(A).]Find an orthonormal basis for ker(A). Questions 2-5: You must show all the steps

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Answered: 4. Consider the matrix A = (a) [ (b) […

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Given the orthonormal basis functions X1 = Po00, X2 = $100, X3 = Po10, X4 = Poo1 which are the eigenfunctions of the ground state and the first excited state of a three-dimensional isotropic harmonic oscillator in Cartesian coordinates, calculate the matrices for the position vector, r %3D %3D
4 0 1] Let A = |-2 1 o). An eigenvalue of A is 3. Find a basis for the corresponding eigenspace, E3. -2
Find a basis for the eigenspace corresponding to the eigenvalue. 1-2 *=[-+²7) A=* A= -4 8 A basis for the eigenspace corresponding to λ = 9 is (Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element. Use a comma to separate answers as needed.)
The linear tranformation L defined by L(p(x)) = -2p' + 9p" maps P4 into P3. (a) Find the matrix representation of L with respect to the ordered bases S = E = {x³, x², x, 1} and F = {x² + x + 1, x + 1, 1} (b) Use Part (a) to find the coordinate vectors of L(p(x)) and L(g(x)) where p(x) = 4x³ + 14x and g(x) = x² + 15. [L(p(x))]F = [L(g(x))] F =
Consider the following matrix: A = 5 4 1. Determine whether A is diagonalizable over C. If so, diagonalize it. 2. Find an invertible matrix Q € Mat22(R), and real numbers r, 0 €R such that 0
1. Answer the following for A = b. Find a basis for the eigenspaces. 1 -3 3 3 3 -5 -3 3 1 ننا تنا
(a) Let V be a finite dimensional vector space and T E L(V). Suppose that the rank of T is 1. Prove that every nonzero vector in the range of T is an eigenvector of Т. (b) Let T E L(C") where n > 1. Suppose that the matrix of T with respect to the standard basis of C" is 1 1 1 1 1 ... 1 ... 1 1 1 ... Find, with explanation, the characteristic polynomial and the minimal polyno- mial of T.
Consider the matrix A 0 1 -1 1 2 0 2 0 0-2 2 0 1 1 (a) Use the Gram-Scmidt method to find an orthogonal basis for Col(A). (b) Use your basis from (a) to find an orthonormal basis for Col(A).
1. a) For the matrix A below, find the eigenvalues, the eigenvectors, a basis B for which the matrix [LA]B is diagonal and the change of basis matrix from the standard basis of R³ to B. 2 0 0 A = -2 45 0 0 1 -4 2 b) For which values of x does the matrix A = have real eigenvalues? X -2 c) Take x = -1 and find the eigenvectors of A.
Linear Algebra
5. Spectral Decomposition
(a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show that (b) Let X = B I3 I1 X2 A c(I+tM)¹ c¹(I - tM) 2 -2 1 1 5 2 -1 2 = 9 e³t -]. 3t f(t) = and Xo = (i) Show that A has a single eigenvalue X. Find A. (ii) Show that A = AI + N where N is a nilpotent matrix. Use this to compute the matrix exponential e¹A (iii) Use the matrix exponential and the method of variation of parameters to solve x' = Ax+ f(t), x(0) = xo.

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4. Consider the matrix A = (a) [ (b) [ 1 -1 0 -1 -1 0 2 -1 2 -1 -1 0 1 -1 ]Find an orthonormal basis for im(A). Find the distance from e3 to im(A). ]Find an orthonormal basis for ker(A).