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LINEAR ALGEBRA
AND ITS APPLICATIONS
THIRD EDITION UPDATE
David C. LayUniversity of Maryland – College Park
STUDY GUIDE
lay_28066_TTL.qxd 5/24/05 12:38 PM Page 1
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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, ortransmitted, in any form or by any means, electronic, mechanical, photocopying, recording, orotherwise, without the prior written permission of the publisher. Printed in the United States ofAmerica.
ISBN 0-321-28066-0
1 2 3 4 5 6 BB 08 07 06 05
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Brief ContentsINTRODUCTION
HOW TO STUDY LINEAR ALGEBRA
C H A P T E R 1 LINEAR EQUATIONS IN LINEAR ALGEBRA
C H A P T E R 2 MATRIX ALGEBRA
C H A P T E R 3 DETERMINANTS
C H A P T E R 4 VECTOR SPACES
C H A P T E R 5 EIGENVALUES AND EIGENVECTORS
C H A P T E R 6 ORTHOGONALITY AND LEAST SQUARES
C H A P T E R 7 SYMMETRIC MATRICES
A P P E N D I C E S TECHNOLOGY INDEX OF PROCEDURES AND TERMS
INTRODUCTION TO MATLAB
NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM
NOTES FOR THE MATHEMATICA COMPUTERALGEBRA SYSTEM
NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORS
NOTES FOR THE HP-48G GRAPHIC CALCULATOR
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ContentsINTRODUCTION viiTechnology Support vii
Review Materials on the Web viii
HOW TO STUDY LINEAR ALGEBRA ixStrategies for Success in Linear Algebra ix
C H A P T E R 1 LINEAR EQUATIONS IN LINEAR ALGEBRA1.1 Systems of Linear Systems 1-1
1.2 Row Reduction and Echelon Forms 1-7
1.3 Vector Equations 1-10
1.4 The Matrix Equation Ax = b 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 1-42
Supplementary Exercises 1-46
Glossary Checklist 1-47
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C H A P T E R 2 MATRIX ALGEBRA2.1 Matrix Operations 2-1
2.2 The Inverse of a Matrix 2-4
2.3 Characterizations of Invertible Matrices 2-9
Expanded Table for the IMT 2-10
Mastering Linear Algebra Concepts: Reviewing and Reflecting
2.4 Partitioned Matrices 2-14
The Principles of Induction 2-20
2.5 Matrix Factorizations 2-20
Permuted LU Factorizations 2-24
2.6 The Leontief Input-Output Model 2-28
2.7 Applications to Computer Graphics 2-30
2.8 Subspaces of Rn 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 2-42
Glossary Checklist 2-42
C H A P T E R 3 DETERMINANTS3.1 Introduction to Determinants 3-1
3.2 Properties of Determinants 3-5
3.3 Cramer’s Rule, Volume and Linear Transformations 3-8
A Geometric Proof 3-12
Glossary Checklist 3-13
C H A P T E R 4 VECTOR SPACES4.1 Vector Spaces and Subspaces 4-1
4.2 Null Spaces, Column Spaces, and Linear Transformations 4-4
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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 4-16
4.6 Rank 4-19
Expanded Table for the IMT 4-21
Mastering Linear Algebra Concepts: Major Review of Key Concepts
4.7 Change of Basis 4-24
4.8 Applications to Difference Equations 4-27
The Casorati Test 4-31
4.9 Applications to Markov Chains 4-32
Glossary Checklist 4-35
C H A P T E R 5 EIGENVALUES AND EIGENVECTORS5.1 Eigenvectors and Eigenvalues 5-1
5.2 The Characteristic Equation 5-5
Factoring a Polynomial 5-8
5.3 Diagonalization 5-9
Mastering Linear Algebra Concepts: Eigenvalue, Eigenvector, Eigenspace
5.4 Eigenvalues and Linear Transformations 5-16
5.5 Complex Eigenvalues 5-19
5.6 Discrete Dynamical Systems 5-22
5.7 Applications to Differential Equations 5-26
5.8 Iterative Estimates for Eigenvalues 5-29
Glossary Checklist 5-34
C H A P T E R 6 ORTHOGONALITY AND LEAST SQUARES6.1 Inner Product, Length, and Orthogonality 6-1
6.2 Orthogonal Sets 6-2
Mastering Linear Algebra Concepts: Orthogonal Basis
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6.3 Orthogonal Projections 6-5
6.4 The Gram-Schmidt Process 6-9
6.5 Least-Squares Problems 6-12
6.6 Applications to Linear Models 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 6-25
Glossary Checklist 6-26
C H A P T E R 7 SYMMETRIC MATRICES7.1 Diagonalization of Symmetric Matrices 7-1
7.2 Quadratic Forms 7-6
Mastering Linear Algebra Concepts: Diagonalization and Quadratic Forms
7.3 Constrained Optimization 7-8
7.4 The Singular Value Decomposition 7-10
Computing an SVD 7-11
7.5 Applications to Image Processing and Statistics 7-16
Supplementary Exercises 7-19
Glossary Checklist 7-19
A P P E N D I C E S TECHNOLOGY INDEX OF PROCEDURES AND TERMS
Technology Index of Procedures and Terms TECH-1
INTRODUCTION TO MATLABGetting Started with Matlab ML-1
Script M-Files ML-3
Index of Matlab Commands ML-5
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NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEMGetting Started with Maple MP-1
Study Guide Notes MP-4
Index of Maple Commands MP-22
NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEMGetting Started with Mathematica MM-1
Study Guide Notes MM-5
Index of Mathematica Commands MM-24
NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORSGetting 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 CALCULATORGetting Started with an HP-48G Calculator HP-1
Study Guide Notes HP-2
Index of HP Calculator Commands HP-18
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