Instructor's Solutions Manual for Elementary Linear Algebra with Applications, 9th Edition

172 Pages·2008·2.45 MB·English

by  Bernard Kolman

Instructor’s Solutions Manual
Elementary Linear
Algebra with
Applications
Ninth Edition
Bernard Kolman
Drexel University
David R. Hill
Temple University
Editorial Director, Computer Science, Engineering, and Advanced Mathematics: Marcia J. Horton
Senior Editor: Holly Stark
Editorial Assistant: Jennifer Lonschein
Senior Managing Editor/Production Editor: Scott Disanno
Art Director: Juan Lo´pez
Cover Designer: Michael Fruhbeis
Art Editor: Thomas Benfatti
Manufacturing Buyer: Lisa McDowell
Marketing Manager: Tim Galligan
Cover Image: (c) William T. Williams, Artist, 1969 Trane, 1969 Acrylic on canvas, 108!! 84!!.
×
Collection of The Studio Museum in Harlem. Gift of Charles Cowles, New York.
c 2008, 2004, 2000, 1996 by Pearson Education, Inc.
"
Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
Earlier editions c 1991, 1986, 1982, by KTI;
"
1977, 1970 by Bernard Kolman
All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in
writing from the publisher.
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
ISBN 0-13-229655-1
Pearson Education, Ltd., London
Pearson Education Australia PTY. Limited, Sydney
Pearson Education Singapore, Pte., Ltd
Pearson Education North Asia Ltd, Hong Kong
Pearson Education Canada, Ltd., Toronto
Pearson Educaci´on de Mexico, S.A. de C.V.
Pearson Education—Japan, Tokyo
Pearson Education Malaysia, Pte. Ltd
Contents
Preface iii
1 Linear Equations and Matrices 1
1.1 Systems of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Algebraic Properties of Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Special Types of Matrices and Partitioned Matrices. . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Matrix Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Computer Graphics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Solving Linear Systems 27
2.1 Echelon Form of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Solving Linear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Elementary Matrices; Finding A 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
−
2.4 Equivalent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 LU-Factorization (Optional). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Determinants 37
3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Cofactor Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Other Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Real Vector Spaces 45
4.1 Vectors in the Plane and in 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Span and Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.8 Coordinates and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.9 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ii CONTENTS
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Inner Product Spaces 71
5.1 Standard Inner Product on R2 and R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Cross Product in R3 (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Inner Product Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Least Squares (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Linear Transformations and Matrices 93
6.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Kernel and Range of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4 Vector Space of Matrices and Vector Space of Linear Transformations (Optional) . . . . . . . 99
6.5 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.6 Introduction to Homogeneous Coordinates (Optional) . . . . . . . . . . . . . . . . . . . . . . 103
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Eigenvalues and Eigenvectors 109
7.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2 Diagonalization and Similar Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3 Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8 Applications of Eigenvalues and Eigenvectors (Optional) 129
8.1 Stable Age Distribution in a Population; Markov Processes . . . . . . . . . . . . . . . . . . . 129
8.2 Spectral Decomposition and Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 130
8.3 Dominant Eigenvalue and Principal Component Analysis . . . . . . . . . . . . . . . . . . . . 130
8.4 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.6 Real Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.7 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.8 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10 MATLAB Exercises 137
Appendix B Complex Numbers 163
B.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.2 Complex Numbers in Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Preface
This manual is to accompany the Ninth Edition of Bernard Kolman and David R.Hill’s Elementary Linear
Algebra with Applications. Answers to all even numbered exercises and detailed solutions to all theoretical
exercises are included. It was prepared by Dennis Kletzing, Stetson University. It contains many of the
solutions found in the Eighth Edition, as well as solutions to new exercises included in the Ninth Edition of
the text.

Chapter 1
Linear Equations and Matrices
Section 1.1, p. 8
2. x=1, y =2, z = 2.
−
4. No solution.
6. x=13+10t, y = 8 8t, t any real number.
− −
8. Inconsistent; no solution.
10. x=2, y = 1.
−
12. No solution.
14. x= 1, y =2, z = 2.
− −
16. (a) For example: s=0, t=0 is one answer.
(b) For example: s=3, t=4 is one answer.
(c) s= t.
2
18. Yes. The trivial solution is always a solution to a homogeneous system.
20. x=1, y =1, z =4.
22. r = 3.
−
24. If x = s , x = s , ..., x = s satisfy each equation of (2) in the original order, then those
1 1 2 2 n n
same numbers satisfy each equation of (2) when the equations are listed with one of the original ones
interchanged, and conversely.
25. If x = s , x = s , ..., x = s is a solution to (2), then the pth and qth equations are satisfied.
1 1 2 2 n n
That is,
a s + +a s =b
p1 1 pn n p
···
a s + +a s =b .
q1 1 qn n q
···
Thus, for any real number r,
(a +ra )s + +(a +ra )s =b +rb .
p1 q1 1 pn qn n p q
···
Then if the qth equation in (2) is replaced by the preceding equation, the values x =s , x =s , ...,
1 1 2 2
x =s are a solution to the new linear system since they satisfy each of the equations.
n n
2 Chapter 1
26. (a) A unique point.
(b) There are infinitely many points.
(c) No points simultaneously lie in all three planes.
C
2
28. No points of intersection: C C C
1 2 1
One point of intersection: C C
1 2
C C
Two points of intersection: 1 2
Infinitely many points of intersection: C1=C2
30. 20 tons of low-sulfur fuel, 20 tons of high-sulfur fuel.
32. 3.2 ounces of food A, 4.2 ounces of food B, and 2 ounces of food C.
34. (a) p(1)= a(1)2+b(1)+c=a+b+c= 5
−
p( 1)=a( 1)2+b( 1)+c=a b+c=1
− − − −
p(2)= a(2)2+b(2)+c=4a+2b+c=7.
(b) a=5, b= 3, c= 7.
− −
Section 1.2, p. 19
0 1 0 0 1 0 1 1 1 1
1 0 1 1 1 1 0 1 0 0
   
2. (a) A= 0 1 0 0 0 (b) A= 1 1 0 1 0 .
   
0 1 0 0 0 1 0 1 0 0
   
1 1 0 0 0 1 0 0 0 0
   
   
4. a=3, b=1, c=8, d= 2.
−
5 5 8
− 7 7
6. (a) C+E =E+C = 4 2 9 . (b) Impossible. (c) − .
  0 1
5 3 4 ’ (
 
9 3 9 0 10 9
− − −
(d) 12 3 15 . (e) 8 1 2 . (f) Impossible.
− − −   − − 
6 3 9 5 4 3
− − − − −
   
1 2 5 4 5
1 2 3 6 10
8. (a) AT = 2 1 , (AT)T = . (b) 5 2 3 . (c) − .
  2 1 4 −  11 17
3 4 ’ ( 8 9 4 ’ (
   
Section 1.3 3
3 4
0 4 17 2
(d) − . (e) 6 3 . (f) .
4 0   16 6
’ ( 9 10 ’− (
 
1 0 1 0 3 0
10. Yes: 2 +1 = .
0 1 0 0 0 2
’ ( ’ ( ’ (
λ 1 2 3
− − −
12. 6 λ+2 3 .
 − − 
5 2 λ 4
− − −
 
14. Because the edges can be traversed in either direction.
x
1
x
2
16. Let x= .  be an n-vector. Then
.
.
 
x 
 n
 
x 0 x +0 x
1 1 1
x 0 x +0 x
2 2 2
x+0= . + .= . = .=x.
. . . .
. . . .
       
x  0 x +0 x 
 n    n   n
       
n m
18. a =(a +a + +a )+(a +a + +a )+ +(a +a + +a )
ij 11 12 1m 21 22 2m n1 n2 nm
··· ··· ··· ···
i=1j=1
))
=(a +a + +a )+(a +a + +a )+ +(a +a + +a )
11 21 n1 12 22 n2 1m 2m nm
m n ··· ··· ··· ···
= a .
ij
j=1i=1
))
n n n n
19. (a) True. (a +1)= a + 1= a +n.
i i i
i=1 i=1 i=1 i=1
) ) ) )
n m n
(b) True. 1 = m=mn.
 
i=1 j=1 i=1
) ) )
 
n m m m m
(c) True. a b =a b +a b + +a b
i j 1 j 2 j n j
   ···
i=1 j=1 j=1 j=1 j=1
) ) ) ) )
m
  
=(a +a + +a ) b
1 2 n j
···
j=1
)
n m m n
= a b = a b
i j i j
. /
i=1 j=1 j=1 i=1
) ) ) )
20. “new salaries”=u+.08u=1.08u.
Section 1.3, p. 30
2. (a) 4. (b) 0. (c) 1. (d) 1.
4. x=5.
4 Chapter 1
6. x= √2, y = 3.
± ±
8. x= 5.
±
10. x= 6, y = 12.
5 5
0 1 1 15 7 14 8 8
− −
12. (a) Impossible. (b) 12 5 17 . (c) 23 5 29 . (d) 14 13 . (e) Impossible.
   −   
19 0 22 13 1 17 13 9
−
     
58 12 28 8 38
14. (a) . (b) Same as (a). (c) .
66 13 34 4 41
’ ( ’ (
28 32 16 8 26
(d) Same as (c). (e) ; same. (f) − − − .
16 18 30 0 31
’ ( ’− − (
1 4 2
−
16. (a) 1. (b) 6. (c) 3 0 1 . (d) 2 8 4 . (e) 10.
− − − 
3 12 6
0 1 − −
 
9 0 3
−
(f) 0 0 0 . (g) Impossible.
 
3 0 1
−
 
18. DI =I D =D.
2 2
0 0
20. .
0 0
’ (
1 0
14 18
22. (a)  . (b)  .
0 3
13 13
   
   
1 2 1 1 2 1
− − − −
24. col (AB)=1 2 +3 4 +2 3 ; col (AB)= 1 2 +2 4 +4 3 .
1 2
      −      
3 0 2 3 0 2
− −
           
26. (a) 5. (b) BAT
−
28. Let A= a be m p and B = b be p n.
ij ij
× ×
(a) Let0the 1ith row of A consist0entir1ely of zeros, so that a = 0 for k = 1,2,...,p. Then the (i,j)
ik
entry in AB is
p
a b =0 for j =1,2,...,n.
ik kj
k=1
)
(b) Let the jth column of A consist entirely of zeros, so that a = 0 for k = 1,2,...,m. Then the
kj
(i,j) entry in BA is
m
b a =0 for i=1,2,...,m.
ik kj
k=1
)
x
2 3 3 1 1 2 3 3 1 1 1 7
− − x
3 0 2 0 3 3 0 2 0 3  2 2
30. (a) 2 3 0 4 0. (b) 2 3 0 4 0x3=−3.
0 0 1 −1 1 0 0 1 −1 1xx45  5
 