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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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
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(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).
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.
-3 (a) Find the coordinate vector of x = 2 with respect to the ordered basis 3 of R3: E= 0 -- {B· Q}· A]} -6 [x] E = (b) Let F₁ be the ordered basis of R2 given by F₁ = 3 and let F2 be the ordered basis given by 2 F2 = {[(3³}]} [6]} 2 Find the transition matrix PF+F₁ such that [x] F₂ = PF₁₂+F₁ [x] F₁ for all x in R²: P =
Suppose TEL(C³) is defined by T(21, 22, 23) = (621 +372 +423,622 +223, 723) (a) Write down the matrix of T with respect to the standard basis. What are the eigenvalues? (b) What are the characteristic polynomial and minimal polynomial of T? (c) What are the Jordan form and Jordan basis for T?

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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).