STUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE

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1 STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson Addison-Wesley. All rights reserved.

2 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 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 BB Copyright 2006 Pearson Addison-Wesley. All rights reserved.

3 Brief Contents INTRODUCTION HOW TO STUDY LINEAR ALGEBRA CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6 CHAPTER 7 APPENDICES LINEAR EQUATIONS IN LINEAR ALGEBRA MATRIX ALGEBRA DETERMINANTS VECTOR SPACES EIGENVALUES AND EIGENVECTORS ORTHOGONALITY AND LEAST SQUARES SYMMETRIC MATRICES 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.

4 Contents INTRODUCTION vii Technology Support vii Review Materials on the Web viii HOW TO STUDY LINEAR ALGEBRA Strategies for Success in Linear Algebra ix ix CHAPTER 1 LINEAR EQUATIONS IN LINEAR ALGEBRA 1.1 Systems of Linear Systems Row Reduction and Echelon Forms Vector Equations The Matrix Equation Ax = b 1-15 Mastering Linear Algebra Concepts: Span 1.5 Solution Sets of Linear Systems Applications of Linear Systems 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 Copyright 2006 Pearson Addison-Wesley. All rights reserved.

5 CHAPTER 2 MATRIX ALGEBRA 2.1 Matrix Operations The Inverse of a Matrix 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 Matrix Factorizations 2-20 Permuted LU Factorizations The Leontief Input-Output Model Applications to Computer Graphics Subspaces of R n 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 CHAPTER 3 DETERMINANTS 3.1 Introduction to Determinants Properties of Determinants Cramer s Rule, Volume and Linear Transformations 3-8 A Geometric Proof 3-12 Glossary Checklist 3-13 CHAPTER 4 VECTOR SPACES 4.1 Vector Spaces and Subspaces Null Spaces, Column Spaces, and Linear Transformations 4-4 Copyright 2006 Pearson Addison-Wesley. All rights reserved.

6 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 The Dimension of a Vector Space Rank 4-19 Expanded Table for the IMT 4-21 Mastering Linear Algebra Concepts: Major Review of Key Concepts 4.7 Change of Basis Applications to Difference Equations 4-27 The Casorati Test Applications to Markov Chains 4-32 Glossary Checklist 4-35 CHAPTER 5 CHAPTER 6 EIGENVALUES AND EIGENVECTORS 5.1 Eigenvectors and Eigenvalues The Characteristic Equation 5-5 Factoring a Polynomial Diagonalization 5-9 Mastering Linear Algebra Concepts: Eigenvalue, Eigenvector, Eigenspace 5.4 Eigenvalues and Linear Transformations Complex Eigenvalues Discrete Dynamical Systems Applications to Differential Equations Iterative Estimates for Eigenvalues 5-29 Glossary Checklist 5-34 ORTHOGONALITY AND LEAST SQUARES 6.1 Inner Product, Length, and Orthogonality Orthogonal Sets 6-2 Mastering Linear Algebra Concepts: Orthogonal Basis Copyright 2006 Pearson Addison-Wesley. All rights reserved.

7 6.3 Orthogonal Projections The Gram-Schmidt Process Least-Squares Problems Applications to Linear Models 6-16 The Geometrey of a Linear Model Inner Product Spaces Applications of Inner Product Spaces 6-22 The Linearity of an Orthoganal Projection 6-25 Glossary Checklist 6-26 CHAPTER 7 SYMMETRIC MATRICES 7.1 Diagonalization of Symmetric Matrices Quadratic Forms 7-6 Mastering Linear Algebra Concepts: Diagonalization and Quadratic Forms 7.3 Constrained Optimization The Singular Value Decomposition 7-10 Computing an SVD Applications to Image Processing and Statistics 7-16 Supplementary Exercises 7-19 Glossary Checklist 7-19 APPENDICES 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-3 Index of Matlab Commands ML-1 ML-5 Copyright 2006 Pearson Addison-Wesley. All rights reserved.

8 NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM Getting Started with Maple Study Guide Notes MP-4 Index of Maple Commands MP-1 MP-22 NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEM Getting Started with Mathematica Study Guide Notes MM-5 Index of Mathematica Commands MM-1 MM-24 NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORS Getting Started with a TI-83+ Calculator Getting Started with a TI-86 Calculator Getting Started with a TI-89 Calculator Study Guide Notes TI-4 Index of TI Calculator Commands TI-29 TI-1 TI-2 TI-3 NOTES FOR THE HP-48G GRAPHIC CALCULATOR Getting Started with an HP-48G Calculator Study Guide Notes HP-2 Index of HP Calculator Commands HP-18 HP-1 Copyright 2006 Pearson Addison-Wesley. All rights reserved.

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