Complete Solution Manual Linear Algebra A Modern Introduction 4th Edition David Poole

Linear Algebra A Modern Introduction 4th Edition David Poole Solutions Manual Contents 1 Vectors 3 1.1 The Geometry and Algebra of Vectors ................................ ................................ ................................ ...... 3 1.2 Length and Angle: The Dot Product ................................ ................................ ................................ . 10 Exploration: Vectors and Geometry ................................ ................................ ................................ .................. 25 1.3 Lines and Planes ................................ ................................ ................................ ................................ ......... 27 Exploration: The Cross Product ................................ ................................ ................................ ................. 41 1.4 Applications ................................ ................................ ................................ ................................ ................. 44 Chapter Review ................................ ................................ ................................ ................................ .................... 48 2 Systems of Linear Equations 53 2.1 Introduction to Systems of Linear Equations ................................ ................................ .......................... 53 2.2 Direct Methods for Solving Linear Systems ................................ ................................ ............................. 58 Exploration: Lies My Computer Told Me ................................ ................................ ................................ .. 75 Exploration: Partial Pivoting ................................ ................................ ................................ ....................... 75 Exploration: An Introduction to the Analysis of Algorithms ................................ ................................ .......... 77 2.3 Spanning Sets and Linear Independence ................................ ................................ ................................ .79 2.4 Applications ................................ ................................ ................................ ................................ ................. 93 2.5 Iterative Methods for Solving Linear Systems ................................ ................................ ...................... 112 Chapter Review ................................ ................................ ................................ ................................ ................. 123 3 Matrices 129 3.1 Matrix Operations ................................ ................................ ................................ ................................ .... 129 3.2 Matrix Algebra ................................ ................................ ................................ ................................ .. 138 3.3 The Inverse of a Matrix ................................ ................................ ................................ .......................... 150 3.4 The LU Factorization ................................ ................................ ................................ ......................... 164 3.5 Subspaces, Basis, Dimension, and Rank ................................ ................................ ................................ 176 3.6 Introduction to Linear Transformations ................................ ................................ ................................ 192 3.7 Applications ................................ ................................ ................................ ................................ .............. 209 Chapter Review ................................ ................................ ................................ ................................ ................. 230 4 Eigenvalues and Eigenvectors 235 4.1 Introduction to Eigenvalues and Eigenvectors ................................ ................................ ..................... 235 4.2 Determinants ................................ ................................ ................................ ................................ ............ 250 Exploration: Geometric Applications of Determinants ................................ ................................ ................ 263 4.3 Eigenvalues and Eigenvectors of n × n Matrices ................................ ................................ ................. 270 4.4 Similarity and Diagonalization ................................ ................................ ................................ ......... 291 4.5 Iterative Methods for Computing Eigenvalues ................................ ................................ ..................... 308 4.6 Applications and the Perron -Frobenius Theorem ................................ ................................ ................ 326 Chapter Review ................................ ................................ ................................ ................................ ................. 365 1 2 CONTENTS 5 Orthogonality 371 5.1 Orthogonality in Rn................................ ................................ ................................ ................................ ................................ .......... 371 5.2 Orthogonal Complements and Orthogonal Projections ................................ ................................ ....... 379 5.3 The Gram -Schmidt Process and the QR Factorization ................................ ................................ .. 388 Exploration: The Modified QR Process ................................ ................................ ................................ ... 398 Exploration: Approximating Eigenvalues with the QR Algorithm ................................ ......................... 402 5.4 Orthogonal Diagonalization of Symmetric Matrices ................................ ................................ ............ 405 5.5 Applications ................................ ................................ ................................ ................................ ............... 417 Chapter Review ................................ ................................ ................................ ................................ .................. 442 6 Vector Spaces 451 6.1 Vector Spaces and Subspaces ................................ ................................ ................................ ................. 451 6.2 Linear Independence, Basis, and Dimension ................................ ................................ ......................... 463 Exploration: Magic Squares ................................ ................................ ................................ ............................. 477 6.3 Change of Basis ................................ ................................ ................................ ................................ ........ 480 6.4 Linear Transformations ................................ ................................ ................................ ............................ 491 6.5 The Kernel and Range of a Linear Transformation ................................ ................................ ............. 498 6.6 The Matrix of a Linear Transformation ................................ ................................ ................................ 507 Exploration: Tiles, Lattices, and the Crystallographic Restriction ................................ ........................ 525 6.7 Applications ................................ ................................ ................................ ................................ ............... 527 Chapter Review ................................ ................................ ................................ ................................ .................. 531 7 Distance and Approximation 537 7.1 Inner Product Spaces ................................ ................................ ................................ ............................... 537 Exploration: Vectors and Matrices with Complex Entries ................................ ................................ ...... 546 Exploration: Geometric Inequalities and Optimization Problems ................................ .............................. 553 7.2 Norms and Distance Functions ................................ ................................ ................................ ............... 556 7.3 Least Squares Approximation ................................ ................................ ................................ ................. 568 7.4 The Singular Value Decomposition ................................ ................................ ................................ ........ 590 7.5 Applications ................................ ................................ ................................ ................................ ............... 614 Chapter Review ................................ ................................ ................................ ................................ .................. 625 8 Codes 633 8.1 Code Vectors ................................ ................................ ................................ ................................ ............. 633 8.2 Error -Correcting Codes ................................ ................................ ................................ ........................... 637 8.3 Dual Codes ................................ ................................ ................................ ................................ ................ 641 8.4 Linear Codes ................................ ................................ ................................ ................................ ............. 647 8.5 The Minimum Distance of a Code ................................ ................................ ................................ ......... 650 −3 0 −3 −3 3 0 −3 3 0 −3 −2 −5 Chapter 1 Vectors 1.1 The Geometry and Algebra of Vectors 1. 2. Since 2 + 3 = 5 , 2 + 2 = 4 , 2 + −2 = 0 , 2 + 3 = 5 , plotting those vectors gives – – – – – 3 (–2, 3) 3 (2, 3) 2 1 –2 –1 1 2 (3, 0) 3 –1 (3, –2) –2 1 2 3 4 5 1 c b 2 a 3 d 4 5 4 CHAPTER 1. VECTORS #−−−−» — − − 2 2 2 2 6 3 2 3 6 6 #−−−−» 3 2 a 1 c b d –1 1 2 3 3. c 4. Since the heads are all at (3, 2, 1), the tails are at 3 0 3 3 3 0 3 1 2 3 −1 4 2 − 2 = 0 , 2 − 2 = 0 , 2 − −2 = 4 , 2 − −1 = 3 . 1 0 1 #−−−−» 1 1 0 1 1 0 1 −2 3 5. The four vectors AB are – – In standard position, the vectors are #−−−−» (a) AB = [4 1, 2 ( 1)] = [3, 3]. #−−−−» (b) AB = [2 − 0, −1 − (−2)] = [2, 1] (c) AB = 1 − 2, 3 − 3 = − 3 , 3 (d) AB = 1 − 1 , 1 − 1 = − 1 , 1 . 2 z 1 b –2 –1 0 1 y 2 0 –1 0 1 a 2 3 x –1 d –2 3 c 2 1 d a 1 2 3 4 1 b 2