Elementary Linear Algebra

the Cornerstone Of Elementary Linear Algebra Is The Authors' Clear, Careful, And Concise Presentation Of Material—written So That Students Can Fully Understand How Mathematics Works. This Program Balances Theory With Examples, Applications, And Geometric Intuition For A Complete, Step-by-step Learning System.the Sixth Edition Incorporates Up-to-date Coverage Of Computer Algebra Systems (maple/matlab/mathematica); Additional Support Is Provided In A Corresponding Technology Guide. Data And Applications Also Reflect Current Statistics And Examples To Engage Students And Demonstrate The Link Between Theory And Practice. Front Cover......Page 1 Title Page......Page 4 Copyright......Page 5 CONTENTS......Page 6 A WORD FROM THE AUTHORS......Page 10 WHAT IS LINEAR ALGEBRA?......Page 18 1.1 Introduction to Systems of Linear Equations......Page 20 1.2 Gaussian Elimination and Gauss-Jordan Elimination......Page 33 1.3 Applications of Systems of Linear Equations......Page 48 Project 1 Graphing Linear Equations......Page 63 Project 2 Underdetermined and Overdetermined Systems of Equations......Page 64 2.1 Operations with Matrices......Page 65 2.2 Properties of Matrix Operations......Page 80 2.3 The Inverse of a Matrix......Page 92 2.4 Elementary Matrices......Page 106 2.5 Applications of Matrix Operations......Page 117 Project 1 Exploring Matrix Multiplication......Page 139 Project 2 Nilpotent Matrices......Page 140 3.1 The Determinant of a Matrix......Page 141 3.2 Evaluation of a Determinant Using Elementary Operations......Page 151 3.3 Properties of Determinants......Page 161 3.4 Introduction to Eigenvalues......Page 171 3.5 Applications of Determinants......Page 177 Project 1 Eigenvalues and Stochastic Matrices......Page 193 Project 2 The Cayley-Hamilton Theorem......Page 194 Cumulative Test for Chapters 1–3......Page 196 4.1 Vectors in R[sup(n)]......Page 198 4.2 Vector Spaces......Page 210 4.3 Subspaces of Vector Spaces......Page 217 4.4 Spanning Sets and Linear Independence......Page 226 4.5 Basis and Dimension......Page 240 4.6 Rank of a Matrix and Systems of Linear Equations......Page 251 4.7 Coordinates and Change of Basis......Page 268 4.8 Applications of Vector Spaces......Page 281 Project 1 Solutions of Linear Systems......Page 294 Project 2 Direct Sum......Page 295 5.1 Length and Dot Product in R[sup(n)]......Page 296 5.2 Inner Product Spaces......Page 311 5.3 Orthonormal Bases: Gram-Schmidt Process......Page 325 5.4 Mathematical Models and Least Squares Analysis......Page 339 5.5 Applications of Inner Product Spaces......Page 355 Project 1 The QR-Factorization......Page 375 Project 2 Orthogonal Matrices and Change of Basis......Page 376 Cumulative Test for Chapters 4 and 5......Page 378 6.1 Introduction to Linear Transformations......Page 380 6.2 The Kernel and Range of a Linear Transformation......Page 393 6.3 Matrices for Linear Transformations......Page 406 6.4 Transition Matrices and Similarity......Page 418 6.5 Applications of Linear Transformations......Page 426 Project 1 Reflections in the Plane (I)......Page 438 Project 2 Reflections in the Plane (II)......Page 439 7.1 Eigenvalues and Eigenvectors......Page 440 7.2 Diagonalization......Page 454 7.3 Symmetric Matrices and Orthogonal Diagonalization......Page 465 7.4 Applications of Eigenvalues and Eigenvectors......Page 477 Project 1 Population Growth and Dynamical Systems (I)......Page 496 Project 2 The Fibonacci Sequence......Page 497 Cumulative Test for Chapters 6 and 7......Page 498 APPENDIX: MATHEMATICAL INDUCTION AND OTHER A1 FORMS OF PROOFS......Page 500 ANSWER KEY......Page 508 INDEX......Page 558