Linear Algebra with Applications (2-Download)
Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Chapter 7.3, Problem 34E

Suppose that B = S − 1 A S for some n × n matrices A, B, and S.
a. Show that if x → is in kerB, then S x → is in kerA.
b. Show that the linear transformation T ( x → ) = S x → from kerB to kerA is an isomorphism.
c. Show that nullity A = nullity B and rank A = rank B.

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Chapter 7 Solutions

Linear Algebra with Applications (2-Download)

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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