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STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE
David C. Lay University of Maryland – College Park
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Reproduced by Pearson Addison-Wesley from electronic files supplied by the author. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley, 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
0-321-28066-0
1 2 3 4 5 6 BB 08 07 06 05
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
Brief Contents INTRODUCTION HOW TO STUDY LINEAR ALGEBRA CHAPTER 1
LINEAR EQUATIONS IN LINEAR ALGEBRA
CHAPTER 2
MATRIX ALGEBRA
CHAPTER 3
DETERMINANTS
CHAPTER 4
VECTOR SPACES
CHAPTER 5
EIGENVALUES AND EIGENVECTORS
CHAPTER 6
ORTHOGONALITY AND LEAST SQUARES
CHAPTER 7
SYMMETRIC MATRICES
APPENDICES
TECHNOLOGY INDEX OF PROCEDURES AND TERMS INTRODUCTION TO MATLAB NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEM NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORS
NOTES FOR THE HP-48G GRAPHIC CALCULATOR Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
Contents INTRODUCTION Technology Support
vii vii
Review Materials on the Web
viii
HOW TO STUDY LINEAR ALGEBRA Strategies for Success in Linear Algebra
CHAPTER
1
ix
ix
LINEAR EQUATIONS IN LINEAR ALGEBRA 1.1
Systems of Linear Systems
1-1
1.2
Row Reduction and Echelon Forms
1.3
Vector Equations
1.4
The Matrix Equation Ax = b
1-7
1-10 1-15
Mastering Linear Algebra Concepts: Span 1.5
Solution Sets of Linear Systems
1-20
1.6
Applications of Linear Systems
1-25
1.7
Linear Independence
1-29
Mastering Linear Algebra Concepts: Linear Independence 1.8
Introduction to Linear Transformations
1-33
Mastering Linear Algebra Concepts: Linear Transformation 1.9
Matrix of a Linear Transformation
1-37
Mastering Linear Algebra Concepts: Existence and Uniqueness 1.10 Linear Models in Business, Science, and Engineering Supplementary Exercises Glossary Checklist
1-46
1-47
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1-42
CHAPTER
2
MATRIX ALGEBRA 2.1
Matrix Operations
2-1
2.2
The Inverse of a Matrix
2.3
Characterizations of Invertible Matrices
2-4
Expanded Table for the IMT
2-9
2-10
Mastering Linear Algebra Concepts: Reviewing and Reflecting 2.4
Partitioned Matrices
2-14
The Principles of Induction 2.5
Matrix Factorizations
2-20
2-20
Permuted LU Factorizations
2-24
2.6
The Leontief Input-Output Model
2.7
Applications to Computer Graphics
2.8
Subspaces of Rn
2-28 2-30
2-33
Mastering Linear Algebra Concepts: Subspace, Column Space, Null Space, Basis 2.9
Dimension and Rank
2-38
Expanded Table for the IMT
2-39
Mastering Linear Algebra Concepts: Dimension and Rank Supplementary Exercises Glossary Checklist
CHAPTER
3
3.1
Introduction to Determinants
3.2
Properties of Determinants
3.3
Cramer’s Rule, Volume and Linear Transformations
Glossary Checklist
4
2-42
DETERMINANTS
A Geometric Proof
CHAPTER
2-42
3-1 3-5 3-8
3-12 3-13
VECTOR SPACES 4.1
Vector Spaces and Subspaces
4-1
4.2
Null Spaces, Column Spaces, and Linear Transformations
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4-4
Mastering Linear Algebra Concepts: Vector Space, Subspace, Col A and Nul A 4.3
Linearly Independent Sets; Bases
4-7
Mastering Linear Algebra Concepts: Basis 4.4
Coordinate Systems
4-11
Isomorphic Vector Spaces
4-12
4.5
The Dimension of a Vector Space
4.6
Rank
4-16
4-19
Expanded Table for the IMT
4-21
Mastering Linear Algebra Concepts: Major Review of Key Concepts 4.7
Change of Basis
4.8
Applications to Difference Equations The Casorati Test
4.9
4-24
4-31
Applications to Markov Chains Glossary Checklist
CHAPTER
5
4-27
4-32
4-35
EIGENVALUES AND EIGENVECTORS 5.1
Eigenvectors and Eigenvalues
5.2
The Characteristic Equation Factoring a Polynomial
5.3
Diagonalization
5-1 5-5
5-8
5-9
Mastering Linear Algebra Concepts: Eigenvalue, Eigenvector, Eigenspace 5.4
Eigenvalues and Linear Transformations
5.5
Complex Eigenvalues
5.6
Discrete Dynamical Systems
5.7
Applications to Differential Equations
5.8
Iterative Estimates for Eigenvalues Glossary Checklist
CHAPTER
6
5-16
5-19 5-22 5-26 5-29
5-34
ORTHOGONALITY AND LEAST SQUARES 6.1
Inner Product, Length, and Orthogonality
6.2
Orthogonal Sets
6-1
6-2
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6.3
Orthogonal Projections
6-5
6.4
The Gram-Schmidt Process
6.5
Least-Squares Problems
6.6
Applications to Linear Models
6-9
6-12 6-16
The Geometrey of a Linear Model
6-19
6.7
Inner Product Spaces
6-20
6.8
Applications of Inner Product Spaces
6-22
The Linearity of an Orthoganal Projection Glossary Checklist CHAPTER
7
6-25
6-26
SYMMETRIC MATRICES 7.1
Diagonalization of Symmetric Matrices
7.2
Quadratic Forms
7-1
7-6
Mastering Linear Algebra Concepts: Diagonalization and Quadratic Forms 7.3
Constrained Optimization
7.4
The Singular Value Decomposition Computing an SVD
7.5
7-10
7-11
Applications to Image Processing and Statistics Supplementary Exercises Glossary Checklist
APPENDICES
7-8
7-16
7-19
7-19
TECHNOLOGY INDEX OF PROCEDURES AND TERMS Technology Index of Procedures and Terms
TECH-1
INTRODUCTION TO MATLAB Getting Started with Matlab Script M-Files
ML-1
ML-3
Index of Matlab Commands
ML-5
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NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM Getting Started with Maple Study Guide Notes
MP-1
MP-4
Index of Maple Commands
MP-22
NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEM Getting Started with Mathematica Study Guide Notes
MM-1
MM-5
Index of Mathematica Commands
MM-24
NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORS Getting Started with a TI-83+ Calculator
TI-1
Getting Started with a TI-86 Calculator
TI-2
Getting Started with a TI-89 Calculator
TI-3
Study Guide Notes
TI-4
Index of TI Calculator Commands
TI-29
NOTES FOR THE HP-48G GRAPHIC CALCULATOR Getting Started with an HP-48G Calculator Study Guide Notes
HP-1
HP-2
Index of HP Calculator Commands
HP-18
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