Linear Algebra With Applications (classic Version)
5th Edition
ISBN: 9780135162972
Author: BRETSCHER, OTTO
Publisher: Pearson Education, Inc.,
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Chapter 7.1, Problem 19E
Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable. 19. Orthogonal projection onto a line L in
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Chapter 7 Solutions
Linear Algebra With Applications (classic Version)
Ch. 7.1 - In Exercises 1 through 4, let A be an invertible...Ch. 7.1 - In Exercises 1 through 4, let A be an invertible...Ch. 7.1 - In Exercises 1 through 4, let A be an invertible...Ch. 7.1 - In Exercises 1 through 4, let A be an invertible...Ch. 7.1 - If a vector is an eigenvector of both A and B, is...Ch. 7.1 - If a vector is an eigenvector of both A and B, is...Ch. 7.1 - If a vector is an eigenvector of the nnmatrixA...Ch. 7.1 - Find all 22 matrix for which e1=[10] is an...Ch. 7.1 - Find all 22 matrix for which e1 is an eigenvector.Ch. 7.1 - Find all 22 matrix for which [12] is an...
Ch. 7.1 - Find all 22 matrix for which [23] is an...Ch. 7.1 - Consider the matrix A=[2034] . Show that 2 and 4...Ch. 7.1 - Show that 4 is an eigenvalue of A=[661513] and...Ch. 7.1 - Find all 44 matrices for which e2 is an...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Arguing geometrically, find all eigenvectors and...Ch. 7.1 - Use matrix products to prove the following: If...Ch. 7.1 - In Exercises 24 through 29, consider a dynamical...Ch. 7.1 - In Exercises 24 through 29, consider a dynamical...Ch. 7.1 - In Exercises 24 through 29, consider a dynamical...Ch. 7.1 - In Exercises 24 through 29, consider a dynamical...Ch. 7.1 - In Exercises 24 through 29, consider a dynamical...Ch. 7.1 - In Exercises 24 through 29, consider a dynamical...Ch. 7.1 - In Exercises 30 through 32, consider the dynamical...Ch. 7.1 - In Exercises 30 through 32, consider the dynamical...Ch. 7.1 - In Exercises 30 through 32, consider the dynamical...Ch. 7.1 - Find a 22 matrix A such that x(t)=[ 2 t 6 t 2 t+ 6...Ch. 7.1 - Suppose is an eigenvector of the nn matrix A,with...Ch. 7.1 - Show that similar matrices have the same...Ch. 7.1 - Find a 22 matrix A such that [31] and [12] are...Ch. 7.1 - Consider the matrix A=[3443] a. Use the geometric...Ch. 7.1 - We are told that [111] is an eigenvector of the...Ch. 7.1 - Find a basis of the linear space V of all 22...Ch. 7.1 - Find a basis of the linear space V of all 22...Ch. 7.1 - Find a basis of the linear space V of all 22...Ch. 7.1 - Find a basis of the linear space V of all 33...Ch. 7.1 - Consider the linear space V of all nn matrices for...Ch. 7.1 - For nn , find the dimension of the space of all nn...Ch. 7.1 - If is any nonzero vector in 2 , what is the...Ch. 7.1 - If is an eigenvector of matrix A with associated...Ch. 7.1 - If is an eigenvector of matrix A, show that is...Ch. 7.1 - If A is a matrix of rank 1, show that any nonzero...Ch. 7.1 - Give an example of a matrix A of rank 1 that fails...Ch. 7.1 - Find an eigenbasis for each of the matrices A in...Ch. 7.1 - Find an eigenbasis for each of the matrices A in...Ch. 7.1 - Find an eigenbasis for each of the matrices A in...Ch. 7.1 - Find an eigenbasis for each of the matrices A in...Ch. 7.1 - Find an eigenbasis for each of the matrices A in...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - Arguing geometrically, find an eigenbasis for each...Ch. 7.1 - In all parts of this problem, let V be the linear...Ch. 7.1 - Consider an nn matrix A. A subspace V of n is...Ch. 7.1 - a. Give an example of a 33 matrix A with as many...Ch. 7.1 - Consider the coyotesroadrunner system discussed...Ch. 7.1 - Two interacting populations of hares and foxes can...Ch. 7.1 - Two interacting populations of coyotes and...Ch. 7.1 - Imagine that you are diabetic and have to pay...Ch. 7.1 - Three holy men (let’s call them Anselm, Benjamin,...Ch. 7.1 - Consider the growth of a lilac bush. The state of...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - For each of the matrices in Exercises 1 through...Ch. 7.2 - Consider a 44 matrix A=[BC0D] , where B, C, and D...Ch. 7.2 - Consider the matrix A=[1k11] , where k is an...Ch. 7.2 - Consider the matrix A=[abbc] , where a, b, and c...Ch. 7.2 - Consider the matrix A=[abba] , where a andb are...Ch. 7.2 - Consider the matrix A=[abba] , where a andb...Ch. 7.2 - True or false? If the determinant of a 22 matrix A...Ch. 7.2 - Ifa 22 matrix A has two distinct eigenvalues 1 and...Ch. 7.2 - Prove the part of Theorem 7.2.8 that concerns the...Ch. 7.2 - Consider an arbitrary nn matrix A. What is...Ch. 7.2 - Suppose matrix A is similar to B. What is the...Ch. 7.2 - Find all eigenvalues of the positive transition...Ch. 7.2 - Consider a positive transition matrix A=[abcd] ,...Ch. 7.2 - Based on your answers in Exercises 24 and 25,...Ch. 7.2 - a. Based on your answers in Exercises 24 and 25,...Ch. 7.2 - Consider the isolated Swiss town of Andelfingen,...Ch. 7.2 - Consider an nn matrix A such that the sum of the...Ch. 7.2 - In all parts of this problem, consider an nn...Ch. 7.2 - Consider a positive transition matrix A. Explain...Ch. 7.2 - Consider the matrix A=[010001k30] wherek is an...Ch. 7.2 - a. Find the characteristic polynomial of the...Ch. 7.2 - Prob. 34ECh. 7.2 - Give an example of a 44 matrix A without real...Ch. 7.2 - For an arbitrary positive integer n, give a...Ch. 7.2 - Prob. 37ECh. 7.2 - IfA isa 22 matrixwith trA=5 and detA=14 ,what are...Ch. 7.2 - Prob. 39ECh. 7.2 - Prob. 40ECh. 7.2 - Prob. 41ECh. 7.2 - Prob. 42ECh. 7.2 - Prob. 43ECh. 7.2 - Prob. 44ECh. 7.2 - For which value of the constant k does the matrix...Ch. 7.2 - In all the parts of this problem, consider a...Ch. 7.2 - Prob. 47ECh. 7.2 - Prob. 48ECh. 7.2 - Prob. 49ECh. 7.2 - Prob. 50ECh. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - Prob. 9ECh. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - Prob. 11ECh. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - Prob. 15ECh. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - Prob. 17ECh. 7.3 - Prob. 18ECh. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - For each of the matrices A in Exercises 1 through...Ch. 7.3 - Find a 22 matrix A for which E1=span[12] and...Ch. 7.3 - Find a 22 matrix A for which E7=2 .Ch. 7.3 - Find all eigenvalues and eigenvectors of A=[1101]...Ch. 7.3 - Find a 22 matrix A for which E1=span[21] is the...Ch. 7.3 - What can you say about the geometric multiplicity...Ch. 7.3 - Show that if a 66 matrix A has a negative...Ch. 7.3 - Consider a 22 matrix A. Suppose that trA=5 and...Ch. 7.3 - Consider the matrix Jn(k)=[000000000k10000k] (with...Ch. 7.3 - Consider a diagonal nn matrix A with rank A=rn ....Ch. 7.3 - Consider an upper triangular nn matrix A with aii0...Ch. 7.3 - Suppose there is an eigenbasis for a matrix A....Ch. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Suppose that B=S1AS for some nn matrices A, B, and...Ch. 7.3 - Is matrix [1203] similar to [3012] ?Ch. 7.3 - Is matrix [0153] similar to [1243] ?Ch. 7.3 - Consider a symmetric nn matrix A. Show that if ...Ch. 7.3 - Consider a rotation T(x)=Ax in 3 . (That is, A is...Ch. 7.3 - Consider a subspace V of n with dim(V)=m . a....Ch. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - Prob. 41ECh. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - Prob. 43ECh. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - Prob. 46ECh. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - Prob. 49ECh. 7.3 - For which values of constants a, b, and c are the...Ch. 7.3 - Prob. 51ECh. 7.3 - Find the characteristic polynomial of the nn...Ch. 7.3 - Prob. 53ECh. 7.3 - Prob. 54ECh. 7.3 - Give an example of a 33 matrix A with nonzero...Ch. 7.3 - Prob. 56ECh. 7.4 - For the matrices A in Exercises 1 through 12, find...Ch. 7.4 - For the matrices A in Exercises 1 through 12, find...Ch. 7.4 - Prob. 3ECh. 7.4 - For the matrices A in Exercises 1 through 12, find...Ch. 7.4 - For the matrices A in Exercises 1 through 12, find...Ch. 7.4 - Prob. 6ECh. 7.4 - Prob. 7ECh. 7.4 - Prob. 8ECh. 7.4 - Prob. 9ECh. 7.4 - Prob. 10ECh. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - Prob. 13ECh. 7.4 - For the matrices A and the vectorsx0in Exercises...Ch. 7.4 - Prob. 15ECh. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - For the matrices A and the vectorsx0in Exercises...Ch. 7.4 - Prob. 19ECh. 7.4 - For the matrices A in Exercises 20 through 24,...Ch. 7.4 - For the matrices A in Exercises 20 through 24,...Ch. 7.4 - Prob. 22ECh. 7.4 - Prob. 23ECh. 7.4 - Prob. 24ECh. 7.4 - Prob. 25ECh. 7.4 - Prob. 26ECh. 7.4 - Prob. 27ECh. 7.4 - Prob. 28ECh. 7.4 - Prob. 29ECh. 7.4 - a. Sketch a phase portrait for the dynamical...Ch. 7.4 - Let x(t) and y(t) be the annual defense budgets of...Ch. 7.4 - Prob. 32ECh. 7.4 - Prob. 33ECh. 7.4 - In an unfortunate accident involving an Austrian...Ch. 7.4 - Prob. 35ECh. 7.4 - A machine contains the grid of wires shown in the...Ch. 7.4 - Prob. 37ECh. 7.4 - Prob. 38ECh. 7.4 - Find all the eigenvalues and “eigenvectors” of the...Ch. 7.4 - Prob. 40ECh. 7.4 - Prob. 41ECh. 7.4 - Prob. 42ECh. 7.4 - Prob. 43ECh. 7.4 - Find all the eigenvalues and “eigenvectors” of the...Ch. 7.4 - Find all the eigenvalues and “eigenvectors” of the...Ch. 7.4 - Prob. 46ECh. 7.4 - Prob. 47ECh. 7.4 - Find all the eigenvalues and “eigenvectors” of the...Ch. 7.4 - Prob. 49ECh. 7.4 - Prob. 50ECh. 7.4 - Find all the eigenvalues and “eigenvectors” of the...Ch. 7.4 - Prob. 52ECh. 7.4 - For a regular transition matrix A, prove the...Ch. 7.4 - Prob. 54ECh. 7.4 - Prob. 55ECh. 7.4 - Prob. 56ECh. 7.4 - Consider an mn matrix A and an nm matrix B. Using...Ch. 7.4 - Prob. 58ECh. 7.4 - Prob. 59ECh. 7.4 - Prob. 60ECh. 7.4 - Prob. 61ECh. 7.4 - Prob. 62ECh. 7.4 - Consider the linear transformation T(f)=f from C...Ch. 7.4 - Prob. 64ECh. 7.4 - Prob. 65ECh. 7.4 - Prob. 66ECh. 7.4 - Consider a 55 matrix A with two distinct...Ch. 7.4 - Prob. 68ECh. 7.4 - We say that two n x n matrices A and B are...Ch. 7.4 - Prob. 70ECh. 7.4 - Prob. 71ECh. 7.4 - Prob. 72ECh. 7.4 - Prove the CayleyHamilton theorem, fA(A)=0 , for...Ch. 7.4 - Prob. 74ECh. 7.5 - Write the complex number z=33i in polar form.Ch. 7.5 - Find all complex numbers z such that z4=1 ....Ch. 7.5 - Prob. 3ECh. 7.5 - Prob. 4ECh. 7.5 - Prob. 5ECh. 7.5 - If z is a nonzero complex number in polar form,...Ch. 7.5 - Prob. 7ECh. 7.5 - Prob. 8ECh. 7.5 - Prob. 9ECh. 7.5 - Prove the fundamental theorem of algebra for cubic...Ch. 7.5 - Prob. 11ECh. 7.5 - Consider a polynomial f() with real coefficients....Ch. 7.5 - For the matrices A listed in Exercises 13 through...Ch. 7.5 - For the matrices A listed in Exercises 13 through...Ch. 7.5 - For the matrices A listed in Exercises 13 through...Ch. 7.5 - For the matrices A listed in Exercises 13 through...Ch. 7.5 - For the matrices A listed in Exercises 13 through...Ch. 7.5 - Prob. 18ECh. 7.5 - Prob. 19ECh. 7.5 - Find all complex eigenvalues of the matrices in...Ch. 7.5 - Find all complex eigenvalues of the matrices in...Ch. 7.5 - Prob. 22ECh. 7.5 - Find all complex eigenvalues of the matrices in...Ch. 7.5 - Find all complex eigenvalues of the matrices in...Ch. 7.5 - Prob. 25ECh. 7.5 - Prob. 26ECh. 7.5 - Suppose a real 33 matrix A has only two distinct...Ch. 7.5 - Suppose a 33 matrix A has the real eigenvalue 2...Ch. 7.5 - Prob. 29ECh. 7.5 - a. If 2i is an eigenvalue of a real 22 matrix A,...Ch. 7.5 - Prob. 31ECh. 7.5 - Prob. 32ECh. 7.5 - Prob. 33ECh. 7.5 - Exercise 33 illustrates how you can use the powers...Ch. 7.5 - Demonstrate the formula trA=1+2+...+n . where the...Ch. 7.5 - In 1990, the population of the African country...Ch. 7.5 - Prob. 37ECh. 7.5 - Prob. 38ECh. 7.5 - Prob. 39ECh. 7.5 - Prob. 40ECh. 7.5 - Prob. 41ECh. 7.5 - Prob. 42ECh. 7.5 - Prob. 43ECh. 7.5 - Prob. 44ECh. 7.5 - Prob. 45ECh. 7.5 - Prob. 46ECh. 7.5 - Prob. 47ECh. 7.5 - Prob. 48ECh. 7.5 - Prob. 49ECh. 7.5 - Prob. 50ECh. 7.5 - Prob. 51ECh. 7.5 - Prob. 52ECh. 7.5 - Prob. 53ECh. 7.5 - Prob. 54ECh. 7.5 - Prob. 55ECh. 7.6 - For the matrices A in Exercises 1 through 10,...Ch. 7.6 - Prob. 2ECh. 7.6 - Prob. 3ECh. 7.6 - Prob. 4ECh. 7.6 - For the matrices A in Exercises 1 through 10,...Ch. 7.6 - Prob. 6ECh. 7.6 - Prob. 7ECh. 7.6 - Prob. 8ECh. 7.6 - For the matrices A in Exercises 1 through 10,...Ch. 7.6 - Prob. 10ECh. 7.6 - Consider the matrices A in Exercises 11 through...Ch. 7.6 - Prob. 12ECh. 7.6 - Prob. 13ECh. 7.6 - Prob. 14ECh. 7.6 - Prob. 15ECh. 7.6 - Prob. 16ECh. 7.6 - For the matrices A in Exercises 17 through 24,...Ch. 7.6 - For the matrices A in Exercises 17 through 24,...Ch. 7.6 - Prob. 19ECh. 7.6 - For the matrices A in Exercises 17 through 24,...Ch. 7.6 - For the matrices A in Exercises 17 through 24,...Ch. 7.6 - For the matrices A in Exercises 17 through 24,...Ch. 7.6 - Prob. 23ECh. 7.6 - Prob. 24ECh. 7.6 - Prob. 25ECh. 7.6 - Prob. 26ECh. 7.6 - Prob. 27ECh. 7.6 - Prob. 28ECh. 7.6 - Consider an invertiblennmatrix A such that the...Ch. 7.6 - Prob. 30ECh. 7.6 - Prob. 31ECh. 7.6 - Prob. 32ECh. 7.6 - Prob. 33ECh. 7.6 - Consider a dynamical system x(t+1)=Ax(t) , whereA...Ch. 7.6 - Prob. 35ECh. 7.6 - Prob. 36ECh. 7.6 - Prob. 37ECh. 7.6 - Prob. 38ECh. 7.6 - Prob. 39ECh. 7.6 - Consider the matrix A=[pqrsqpsrrspqsrqp] , wherep,...Ch. 7.6 - Prob. 41ECh. 7.6 - Prob. 42ECh. 7 - If 0 is an eigenvalue of a matrix A, then detA=0 .Ch. 7 - Prob. 2ECh. 7 - Prob. 3ECh. 7 - Prob. 4ECh. 7 - The algebraic multiplicity of an eigenvalue cannot...Ch. 7 - Prob. 6ECh. 7 - Prob. 7ECh. 7 - Prob. 8ECh. 7 - There exists a diagonalizable 55 matrix with only...Ch. 7 - Prob. 10ECh. 7 - Prob. 11ECh. 7 - Prob. 12ECh. 7 - Prob. 13ECh. 7 - If Ais a noninvertible nn matrix, then the...Ch. 7 - If matrix A is diagonalizable, then its transpose...Ch. 7 - Prob. 16ECh. 7 - Prob. 17ECh. 7 - If A andB are nn matrices, if is an eigenvalue...Ch. 7 - Prob. 19ECh. 7 - Prob. 20ECh. 7 - Prob. 21ECh. 7 - Prob. 22ECh. 7 - Prob. 23ECh. 7 - Prob. 24ECh. 7 - Prob. 25ECh. 7 - Prob. 26ECh. 7 - Prob. 27ECh. 7 - Prob. 28ECh. 7 - Prob. 29ECh. 7 - Prob. 30ECh. 7 - Prob. 31ECh. 7 - If a 44 matrix A is diagonalizable, then the...Ch. 7 - Prob. 33ECh. 7 - Prob. 34ECh. 7 - Prob. 35ECh. 7 - Prob. 36ECh. 7 - Prob. 37ECh. 7 - Prob. 38ECh. 7 - IfAisa22 matrixsuch that trA=1 and detA=6 , then A...Ch. 7 - If a matrix is diagonalizable, then the algebraic...Ch. 7 - Prob. 41ECh. 7 - Prob. 42ECh. 7 - Prob. 43ECh. 7 - Prob. 44ECh. 7 - Prob. 45ECh. 7 - Prob. 46ECh. 7 - Prob. 47ECh. 7 - Prob. 48ECh. 7 - Prob. 49ECh. 7 - Prob. 50ECh. 7 - Prob. 51ECh. 7 - Prob. 52ECh. 7 - Prob. 53ECh. 7 - Prob. 54ECh. 7 - Prob. 55ECh. 7 - Prob. 56ECh. 7 - Prob. 57ECh. 7 - Prob. 58E
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