INSTRUCTOR’S
SOLUTIONS MANUAL
LINEAR A LGEBRA
WITH A PPLICATIONS
NINTH EDITION
Steven J. Leon
University of Massachusetts, Dartmouth
Boston Columbus Indianapolis New York San Francisco
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
,The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2015, 2010, 2006 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-98305-3
ISBN-10: 0-321-98305-X
1 2 3 4 5 6 OPM 17 16 15 14
www.pearsonhighered.com
, Contents
Preface v
1 Matrices and Systems of Equations 1
1 Systems of Linear Equations 1
2 Row Echelon Form 2
3 Matrix Arithmetic 3
4 Matrix Algebra 6
5 Elementary Matrices 12
6 Partitioned Matrices 17
MATLAB Exercises 20
Chapter Test A 22
Chapter Test B 24
2 Determinants 27
1 The Determinant of a Matrix 27
2 Properties of Determinants 30
3 Additional Topics and Applications 33
MATLAB Exercises 35
Chapter Test A 35
Chapter Test B 36
3 Vector Spaces 38
1 Definition and Examples 38
2 Subspaces 42
3 Linear Independence 47
4 Basis and Dimension 50
5 Change of Basis 52
6 Row Space and Column Space 52
MATLAB Exercises 59
Chapter Test A 60
Chapter Test B 62
4 Linear Transformations 66
1 Definition and Examples 66
2 Matrix Representations of Linear Transformations 69
3 Similarity 71
MATLAB Exercise 72
iii
Copyright © 2015 Pearson Education, Inc.
, iv Contents
Chapter Test A 73
Chapter Test B 74
5 Orthogonality 76
1 The Scalar product in Rn 76
2 Orthogonal Subspaces 78
3 Least Squares Problems 81
4 Inner Product Spaces 85
5 Orthonormal Sets 90
6 The Gram-Schmidt Process 98
7 Orthogonal Polynomials 100
MATLAB Exercises 103
Chapter Test A 104
Chapter Test B 105
6 Eigenvalues 109
1 Eigenvalues and Eigenvectors 109
2 Systems of Linear Differential Equations 114
3 Diagonalization 115
4 Hermitian Matrices 123
5 Singular Value Decomposition 130
6 Quadratic Forms 132
7 Positive Definite Matrices 135
8 Nonnegative Matrices 138
MATLAB Exercises 140
Chapter Test A 144
Chapter Test B 145
7 Numerical Linear Algebra 149
1 Floating-Point Numbers 149
2 Gaussian Elimination 150
3 Pivoting Strategies 151
4 Matrix Norms and Condition Numbers 152
5 Orthogonal Transformations 162
6 The Eigenvalue Problem 164
7 Least Squares Problems 168
MATLAB Exercises 171
Chapter Test A 172
Chapter Test B 173
Copyright © 2015 Pearson Education, Inc.
SOLUTIONS MANUAL
LINEAR A LGEBRA
WITH A PPLICATIONS
NINTH EDITION
Steven J. Leon
University of Massachusetts, Dartmouth
Boston Columbus Indianapolis New York San Francisco
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
,The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2015, 2010, 2006 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-98305-3
ISBN-10: 0-321-98305-X
1 2 3 4 5 6 OPM 17 16 15 14
www.pearsonhighered.com
, Contents
Preface v
1 Matrices and Systems of Equations 1
1 Systems of Linear Equations 1
2 Row Echelon Form 2
3 Matrix Arithmetic 3
4 Matrix Algebra 6
5 Elementary Matrices 12
6 Partitioned Matrices 17
MATLAB Exercises 20
Chapter Test A 22
Chapter Test B 24
2 Determinants 27
1 The Determinant of a Matrix 27
2 Properties of Determinants 30
3 Additional Topics and Applications 33
MATLAB Exercises 35
Chapter Test A 35
Chapter Test B 36
3 Vector Spaces 38
1 Definition and Examples 38
2 Subspaces 42
3 Linear Independence 47
4 Basis and Dimension 50
5 Change of Basis 52
6 Row Space and Column Space 52
MATLAB Exercises 59
Chapter Test A 60
Chapter Test B 62
4 Linear Transformations 66
1 Definition and Examples 66
2 Matrix Representations of Linear Transformations 69
3 Similarity 71
MATLAB Exercise 72
iii
Copyright © 2015 Pearson Education, Inc.
, iv Contents
Chapter Test A 73
Chapter Test B 74
5 Orthogonality 76
1 The Scalar product in Rn 76
2 Orthogonal Subspaces 78
3 Least Squares Problems 81
4 Inner Product Spaces 85
5 Orthonormal Sets 90
6 The Gram-Schmidt Process 98
7 Orthogonal Polynomials 100
MATLAB Exercises 103
Chapter Test A 104
Chapter Test B 105
6 Eigenvalues 109
1 Eigenvalues and Eigenvectors 109
2 Systems of Linear Differential Equations 114
3 Diagonalization 115
4 Hermitian Matrices 123
5 Singular Value Decomposition 130
6 Quadratic Forms 132
7 Positive Definite Matrices 135
8 Nonnegative Matrices 138
MATLAB Exercises 140
Chapter Test A 144
Chapter Test B 145
7 Numerical Linear Algebra 149
1 Floating-Point Numbers 149
2 Gaussian Elimination 150
3 Pivoting Strategies 151
4 Matrix Norms and Condition Numbers 152
5 Orthogonal Transformations 162
6 The Eigenvalue Problem 164
7 Least Squares Problems 168
MATLAB Exercises 171
Chapter Test A 172
Chapter Test B 173
Copyright © 2015 Pearson Education, Inc.
