Differential Equations and Linear Algebra (4th Edition)
4th Edition
ISBN: 9780321964670
Author: Stephen W. Goode, Scott A. Annin
Publisher: PEARSON
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Textbook Question
Chapter 9.4, Problem 24P
Let
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Consider the function T : P2(R)
→ P2 (R) given by T(a+ bx + cx²) = (a – b+ c) + bx + (-b+2c)x²
Find eigenvalues of T and state their algebraic multiplicity.
Let A be a 2×2 non defective matrix.If very solution to x′= Ax satisfies (9.4.6), prove that all eigenvalues of A have negative real part.
Let
-
Q(x) = 35x²7x27x3 + 18x1x2 + 18x1x3 − 18x2x3.
Find a unit vector x in R³ whose first entry is positive at which Q(x) is minimized, subject to xx = 1. [Hint: The
eigenvalues of the matrix of the quadratic form Q are 2, -19, and 38.]
Chapter 9 Solutions
Differential Equations and Linear Algebra (4th Edition)
Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 5TFRCh. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 7TFRCh. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 9TFRCh. 9.1 - Prob. 10TFR
Ch. 9.1 - Problems For problems 1-8, solve the given system...Ch. 9.1 - Problems For problems 1-8, solve the given system...Ch. 9.1 - Problems For problems 1-8, solve the given system...Ch. 9.1 - Prob. 4PCh. 9.1 - Problems For problems 1-8, solve the given system...Ch. 9.1 - Prob. 6PCh. 9.1 - Problems For problems 1-8, solve the given system...Ch. 9.1 - Problems For problems 1-8, solve the given system...Ch. 9.1 - Problems For Problems 9-11, solve the given...Ch. 9.1 - Problems For Problem 9-11, solve the given...Ch. 9.1 - Problems For Problem 9-11, solve the given...Ch. 9.1 - Problems For Problems 12-14, solve the given...Ch. 9.1 - Problems For Problems 12-14, solve the given...Ch. 9.1 - Prob. 14PCh. 9.1 - Problems For Problem 15-16, convert the given...Ch. 9.1 - Problems For Problem 15-16, convert the given...Ch. 9.1 - Prob. 17PCh. 9.1 - For Problems 17-19, convert the given linear...Ch. 9.1 - Prob. 19PCh. 9.1 - Prob. 20PCh. 9.1 - Solve the initial-value problem:...Ch. 9.2 - Prob. 1TFRCh. 9.2 - True-False Review For Questions a-g, decide if the...Ch. 9.2 - Prob. 3TFRCh. 9.2 - True-False Review For Questions a-g, decide if the...Ch. 9.2 - Prob. 5TFRCh. 9.2 - Prob. 6TFRCh. 9.2 - Prob. 7TFRCh. 9.2 - Problems
For Problems 1-5, show that the given...Ch. 9.2 - Problems
For Problems 1-5, show that the given...Ch. 9.2 - Problems
For Problems 1-5, show that the given...Ch. 9.2 - Problems
For Problems 1-5, show that the given...Ch. 9.2 - Problems
For Problems 1-5, show that the given...Ch. 9.2 - Problems
For Problems 6-9, show that the given...Ch. 9.2 - Problems
For Problems 6-9, show that the given...Ch. 9.2 - Problems
For Problems 6-9, show that the given...Ch. 9.2 - Problems
For Problems 6-9, show that the given...Ch. 9.2 - Prove that Vn(I) is a vector space.Ch. 9.2 - Problems
Let A(t) be an nn matrix function....Ch. 9.2 - If A=[2413], determine two linearly independent...Ch. 9.2 - Consider the differential equation...Ch. 9.3 - True-False Review For Questions a-d, decide if the...Ch. 9.3 - Prob. 2TFRCh. 9.3 - Prob. 3TFRCh. 9.3 - Prob. 4TFRCh. 9.3 - For Problems 1-7, show that the given functions...Ch. 9.3 - For Problems 1-7, show that the given functions...Ch. 9.3 - For Problems 17, show that the given function are...Ch. 9.3 - For Problems 1-7, show that the given functions...Ch. 9.3 - For Problems 1-7, show that the given functions...Ch. 9.3 - For Problems 1-7, show that the given functions...Ch. 9.3 - For Problems 1-7, show that the given functions...Ch. 9.3 - Prob. 8PCh. 9.3 - Prob. 9PCh. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 116, determine the general solution...Ch. 9.4 - For Problems 1-16, determine the general solution...Ch. 9.4 - For Problems 1-16, determine the general solution...Ch. 9.4 - For Problems 1-16, determine the general solution...Ch. 9.4 - For Problems 1-16, determine the general solution...Ch. 9.4 - For Problems 1-16, determine the general solution...Ch. 9.4 - For Problems 1719, solve the initial-value problem...Ch. 9.4 - Prob. 18PCh. 9.4 - For Problems 1719, solve the initial-value problem...Ch. 9.4 - Prob. 20PCh. 9.4 - Consider the differential equation...Ch. 9.4 - Prob. 22PCh. 9.4 - Prob. 23PCh. 9.4 - Let A be a 22 nondefective matrix. If every...Ch. 9.4 - Prob. 25PCh. 9.4 - Prob. 26PCh. 9.4 - Prob. 27PCh. 9.4 - The motion of a certain physical system is...Ch. 9.5 - True-False Review For Questions a-d, decide if the...Ch. 9.5 - Prob. 2TFRCh. 9.5 - True-False Review For Questions a-d, decide if the...Ch. 9.5 - Prob. 4TFRCh. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - For Problem 114, determine the general solution to...Ch. 9.5 - Prob. 12PCh. 9.5 - Prob. 14PCh. 9.5 - For Problem 1516, solve the initial-value problem....Ch. 9.5 - For Problem 1516, solve the initial-value problem....Ch. 9.5 - Show that if the vector differential equation x=Ax...Ch. 9.5 - Prob. 18PCh. 9.5 - Prob. 19PCh. 9.5 - Prob. 20PCh. 9.6 - True-False Review For Questions (a)(f), decide if...Ch. 9.6 - Prob. 2TFRCh. 9.6 - Prob. 3TFRCh. 9.6 - Prob. 4TFRCh. 9.6 - True-False Review For Questions (a)(f), decide if...Ch. 9.6 - Prob. 6TFRCh. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems For Problems 19, use the variation of...Ch. 9.6 - Problems Let X(t) be a fundamental matric for the...Ch. 9.6 - Problems Consider the nonhomogeneous system...Ch. 9.7 - True-False Review For Questions a-e, decide if the...Ch. 9.7 - Prob. 2TFRCh. 9.7 - True-False Review For Questions a-e, decide if the...Ch. 9.7 - Prob. 4TFRCh. 9.7 - True-False Review For Questions a-e, decide if the...Ch. 9.7 - Derive the eigenvalues and eigenvectors given in...Ch. 9.7 - Prob. 2PCh. 9.7 - Prob. 3PCh. 9.7 - Determine the general motion of the coupled...Ch. 9.7 - Prob. 5PCh. 9.7 - Prob. 6PCh. 9.7 - Prob. 7PCh. 9.7 - Prob. 8PCh. 9.7 - Prob. 9PCh. 9.7 - Prob. 11PCh. 9.8 - Prob. 1TFRCh. 9.8 - Prob. 2TFRCh. 9.8 - Prob. 3TFRCh. 9.8 - Prob. 4TFRCh. 9.8 - Prob. 5TFRCh. 9.8 - Prob. 6TFRCh. 9.8 - Prob. 1PCh. 9.8 - Prob. 2PCh. 9.8 - Prob. 3PCh. 9.8 - Prob. 4PCh. 9.8 - Prob. 5PCh. 9.8 - Prob. 8PCh. 9.8 - Prob. 9PCh. 9.8 - Prob. 10PCh. 9.8 - Prob. 11PCh. 9.9 - True-False Review For Questions a-f, decide if the...Ch. 9.9 - True-False Review For Questions a-f, decide if the...Ch. 9.9 - Prob. 3TFRCh. 9.9 - Prob. 4TFRCh. 9.9 - Prob. 5TFRCh. 9.9 - Prob. 6TFRCh. 9.9 - Prob. 1PCh. 9.9 - Prob. 2PCh. 9.9 - Prob. 3PCh. 9.9 - Prob. 4PCh. 9.9 - Prob. 5PCh. 9.9 - Prob. 6PCh. 9.9 - Prob. 7PCh. 9.9 - Prob. 8PCh. 9.9 - For Problems 5-20, characterize the equilibrium...Ch. 9.9 - Prob. 10PCh. 9.9 - Prob. 11PCh. 9.9 - Prob. 12PCh. 9.9 - Prob. 13PCh. 9.9 - Prob. 14PCh. 9.9 - For Problems 5-20, characterize the equilibrium...Ch. 9.9 - Prob. 16PCh. 9.9 - For Problems 5-20, characterize the equilibrium...Ch. 9.9 - For Problems 5-20, characterize the equilibrium...Ch. 9.9 - For Problems 5-20, characterize the equilibrium...Ch. 9.9 - Prob. 20PCh. 9.9 - Prob. 21PCh. 9.9 - Prob. 22PCh. 9.9 - For Problems 22-25, convert the given differential...Ch. 9.9 - Prob. 26PCh. 9.9 - Prob. 27PCh. 9.10 - For Questions a-e, decide if the given statement...Ch. 9.10 - Prob. 1PCh. 9.10 - Prob. 2PCh. 9.10 - Prob. 3PCh. 9.10 - Problems For Problems 1-9, determine all...Ch. 9.10 - Problems For Problems 1-9, determine all...Ch. 9.10 - Problems For Problems 1-9, determine all...Ch. 9.10 - Prob. 7PCh. 9.10 - Prob. 8PCh. 9.10 - Prob. 9PCh. 9.10 - Prob. 10PCh. 9.10 - Prob. 11PCh. 9.10 - Prob. 12PCh. 9.10 - Prob. 13PCh. 9.10 - Prob. 14PCh. 9.10 - Prob. 15PCh. 9.10 - Prob. 16PCh. 9.10 - Problem For Problems 13-18, sketch the phase...Ch. 9.10 - Prob. 18PCh. 9.10 - Prob. 19PCh. 9.10 - Prob. 20PCh. 9.10 - Problems Consider the differential equation...Ch. 9.11 - Additional Problems Verify that x1(t)=[et2t1] and...Ch. 9.11 - Prob. 2APCh. 9.11 - Prob. 3APCh. 9.11 - Prob. 4APCh. 9.11 - Prob. 5APCh. 9.11 - Prob. 6APCh. 9.11 - Prob. 7APCh. 9.11 - Prob. 8APCh. 9.11 - Additional Problems For Problems 3-24, determine...Ch. 9.11 - ADDITIONAL PROBLEMS For Problems 3-24, determine...Ch. 9.11 - Prob. 11APCh. 9.11 - Prob. 12APCh. 9.11 - Prob. 13APCh. 9.11 - Prob. 14APCh. 9.11 - For Problems 324, determine the general solution...Ch. 9.11 - Prob. 16APCh. 9.11 - Prob. 17APCh. 9.11 - Prob. 18APCh. 9.11 - Prob. 19APCh. 9.11 - Prob. 20APCh. 9.11 - Prob. 21APCh. 9.11 - Prob. 22APCh. 9.11 - Prob. 23APCh. 9.11 - Prob. 24APCh. 9.11 - Prob. 25APCh. 9.11 - For Problems 2529, use the variation-of-parameters...Ch. 9.11 - For Problems 2529, use the variation-of-parameters...Ch. 9.11 - Prob. 28APCh. 9.11 - Prob. 29APCh. 9.11 - True or False: If X(t) is a fundamental matrix for...Ch. 9.11 - Prob. 31APCh. 9.11 - Prob. 32APCh. 9.11 - Consider the differential equation...Ch. 9.11 - Prob. 34APCh. 9.11 - For Problems 34-41, characterize the equilibrium...Ch. 9.11 - Prob. 36APCh. 9.11 - Prob. 37APCh. 9.11 - Prob. 38APCh. 9.11 - Prob. 39APCh. 9.11 - Prob. 40APCh. 9.11 - Prob. 41APCh. 9.11 - Prob. 42APCh. 9.11 - Prob. 43AP
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- Define T:P2P2 by T(a0+a1x+a2x2)=(2a0+a1a2)+(a1+2a2)xa2x2. Find the eigenvalues and the eigenvectors of T relative to the standard basis {1,x,x2}.arrow_forwardX1(t) x2(t) a Suppose x'(t) = Ax(t) where A = ), a, b, c, d e R and æ(t) Use the d definition of linear independence to justify each of the following. Recall that if v E R" = 0, then each component of v must equal 0. (a) Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution x(t) are linearly independent. (b) Suppose there is only one eigenvalue of A. Show that the terms in the solution x(t) are linearly independent. (Hint: There are two casees here linearly independent eigenvectors associated with that eigenvalue and the second case is when we cannot find two linearly independent eigenvalues) - one case when there are two (c) Suppose the eigenvalues of A are complex conjugates. Show that the terms in the solution x(t) are linearly independent.arrow_forwardLet f(x) = xT Ax be a quadratic form with associated n X n symmetric matrix A. Let the eigenvalues of A be λ1 >= λ2>= ···>= λn Then the following are true, subject to the constraint II x|| = 1: Prove The minimum value of f(x) is λn, and it occurs when x is a unit eigenvector corresponding to λnarrow_forward
- Let Q(x) = 143x² – 34x2 – 34x² – 36x1x2 – 36x1x3 – 72x2x3. - - T Find a unit vector x in R³ whose first entry is positive at which Q(x) is minimized, subject to x¹x = 1. [Hint: The eigenvalues of the matrix of the quadratic form Q are 2, -73, and 146.] Enter the components of x (in order) into the answer box below, separated with commas.arrow_forwardAssume v (0) = (1 1 0) T, iterate the matrix until the error value E|<0.005 by using (a) power method. Find the dominant eigenvalue, Ajargest in absolute value and show the eigenvector, vi of matrix A. Give your answer to three decimal places. (3 1 2) A= | 1 0 1 2 1 3arrow_forwardLet T: R5 → R5 be a linear operator whose characteristic polynomial is pT(x) = (x−3)3(x−5)2. Let V3 and V5 be the eigenspaces associated with eigenvalues 3 and 5, respectively. Judge each item below as true or false. (A) If dimV3 = dimV5, then T is diagonalizable (B) If T is diagonalizable, then there is a basis B of R5 formed by eigenvectors such that det([T]BB) = 675. (C) If V3 is generated by v1,v2 ∈ R5 and V5 is generated by u1, u2 ∈ R5, then T is not diagonalizable.arrow_forward
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