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 AddisonWesley. All rights reserved.
2 Reproduced by Pearson AddisonWesley from electronic files supplied by the author. Copyright 2006 Pearson Education, Inc. Publishing as Pearson AddisonWesley, 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 AddisonWesley. 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 TI83+/86/89 GRAPHIC CALCULATORS NOTES FOR THE HP48G GRAPHIC CALCULATOR Copyright 2006 Pearson AddisonWesley. 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 115 Mastering Linear Algebra Concepts: Span 1.5 Solution Sets of Linear Systems Applications of Linear Systems Linear Independence 129 Mastering Linear Algebra Concepts: Linear Independence 1.8 Introduction to Linear Transformations 133 Mastering Linear Algebra Concepts: Linear Transformation 1.9 Matrix of a Linear Transformation 137 Mastering Linear Algebra Concepts: Existence and Uniqueness 1.10 Linear Models in Business, Science, and Engineering 142 Supplementary Exercises 146 Glossary Checklist 147 Copyright 2006 Pearson AddisonWesley. All rights reserved.
5 CHAPTER 2 MATRIX ALGEBRA 2.1 Matrix Operations The Inverse of a Matrix Characterizations of Invertible Matrices 29 Expanded Table for the IMT 210 Mastering Linear Algebra Concepts: Reviewing and Reflecting 2.4 Partitioned Matrices 214 The Principles of Induction Matrix Factorizations 220 Permuted LU Factorizations The Leontief InputOutput Model Applications to Computer Graphics Subspaces of R n 233 Mastering Linear Algebra Concepts: Subspace, Column Space, Null Space, Basis 2.9 Dimension and Rank 238 Expanded Table for the IMT 239 Mastering Linear Algebra Concepts: Dimension and Rank Supplementary Exercises 242 Glossary Checklist 242 CHAPTER 3 DETERMINANTS 3.1 Introduction to Determinants Properties of Determinants Cramer s Rule, Volume and Linear Transformations 38 A Geometric Proof 312 Glossary Checklist 313 CHAPTER 4 VECTOR SPACES 4.1 Vector Spaces and Subspaces Null Spaces, Column Spaces, and Linear Transformations 44 Copyright 2006 Pearson AddisonWesley. All rights reserved.
6 Mastering Linear Algebra Concepts: Vector Space, Subspace, Col A and Nul A 4.3 Linearly Independent Sets; Bases 47 Mastering Linear Algebra Concepts: Basis 4.4 Coordinate Systems 411 Isomorphic Vector Spaces The Dimension of a Vector Space Rank 419 Expanded Table for the IMT 421 Mastering Linear Algebra Concepts: Major Review of Key Concepts 4.7 Change of Basis Applications to Difference Equations 427 The Casorati Test Applications to Markov Chains 432 Glossary Checklist 435 CHAPTER 5 CHAPTER 6 EIGENVALUES AND EIGENVECTORS 5.1 Eigenvectors and Eigenvalues The Characteristic Equation 55 Factoring a Polynomial Diagonalization 59 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 529 Glossary Checklist 534 ORTHOGONALITY AND LEAST SQUARES 6.1 Inner Product, Length, and Orthogonality Orthogonal Sets 62 Mastering Linear Algebra Concepts: Orthogonal Basis Copyright 2006 Pearson AddisonWesley. All rights reserved.
7 6.3 Orthogonal Projections The GramSchmidt Process LeastSquares Problems Applications to Linear Models 616 The Geometrey of a Linear Model Inner Product Spaces Applications of Inner Product Spaces 622 The Linearity of an Orthoganal Projection 625 Glossary Checklist 626 CHAPTER 7 SYMMETRIC MATRICES 7.1 Diagonalization of Symmetric Matrices Quadratic Forms 76 Mastering Linear Algebra Concepts: Diagonalization and Quadratic Forms 7.3 Constrained Optimization The Singular Value Decomposition 710 Computing an SVD Applications to Image Processing and Statistics 716 Supplementary Exercises 719 Glossary Checklist 719 APPENDICES TECHNOLOGY INDEX OF PROCEDURES AND TERMS Technology Index of Procedures and Terms TECH1 INTRODUCTION TO MATLAB Getting Started with Matlab Script MFiles ML3 Index of Matlab Commands ML1 ML5 Copyright 2006 Pearson AddisonWesley. All rights reserved.
8 NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM Getting Started with Maple Study Guide Notes MP4 Index of Maple Commands MP1 MP22 NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEM Getting Started with Mathematica Study Guide Notes MM5 Index of Mathematica Commands MM1 MM24 NOTES FOR THE TI83+/86/89 GRAPHIC CALCULATORS Getting Started with a TI83+ Calculator Getting Started with a TI86 Calculator Getting Started with a TI89 Calculator Study Guide Notes TI4 Index of TI Calculator Commands TI29 TI1 TI2 TI3 NOTES FOR THE HP48G GRAPHIC CALCULATOR Getting Started with an HP48G Calculator Study Guide Notes HP2 Index of HP Calculator Commands HP18 HP1 Copyright 2006 Pearson AddisonWesley. All rights reserved.
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