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Elementary Linear Algebra

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Elementary Linear Algebra Elementary Linear Algebra Kuttler March 24, 2009 2 Contents 1 Introduction 7 2 Fn 9 2.0.1 Outcomes .................................. 9 2.1 Algebra in Fn .................................... 11 2.2 Geometric Meaning Of Vectors .......................... 12 2.3 Geometric Meaning Of Vector Addition ..................... 12 2.4 Distance Between Points In Rn Length Of A Vector . 14 2.5 Geometric Meaning Of Scalar Multiplication . 17 2.6 Vectors And Physics ................................ 19 2.7 Exercises With Answers .............................. 23 3 Systems Of Equations 25 3.0.1 Outcomes .................................. 25 3.1 Systems Of Equations, Geometric Interpretations . 25 3.2 Systems Of Equations, Algebraic Procedures . 28 3.2.1 Elementary Operations .......................... 28 3.2.2 Gauss Elimination ............................. 30 4 Matrices 41 4.0.3 Outcomes .................................. 41 4.1 Matrix Arithmetic ................................. 41 4.1.1 Addition And Scalar Multiplication Of Matrices . 41 4.1.2 Multiplication Of Matrices ........................ 44 4.1.3 The ijth Entry Of A Product ....................... 47 4.1.4 Properties Of Matrix Multiplication ................... 49 4.1.5 The Transpose ............................... 50 4.1.6 The Identity And Inverses ......................... 51 4.1.7 Finding The Inverse Of A Matrix ..................... 53 5 Vector Products 59 5.0.8 Outcomes .................................. 59 5.1 The Dot Product .................................. 59 5.2 The Geometric Signi¯cance Of The Dot Product . 61 5.2.1 The Angle Between Two Vectors ..................... 61 5.2.2 Work And Projections ........................... 63 5.2.3 The Dot Product And Distance In Cn . 65 5.3 Exercises With Answers .............................. 68 5.4 The Cross Product ................................. 69 5.4.1 The Distributive Law For The Cross Product . 72 3 4 CONTENTS 5.4.2 The Box Product .............................. 74 5.4.3 A Proof Of The Distributive Law ..................... 75 6 Determinants 77 6.0.4 Outcomes .................................. 77 6.1 Basic Techniques And Properties ......................... 77 6.1.1 Cofactors And 2 £ 2 Determinants .................... 77 6.1.2 The Determinant Of A Triangular Matrix . 81 6.1.3 Properties Of Determinants ........................ 82 6.1.4 Finding Determinants Using Row Operations . 83 6.2 Applications ..................................... 85 6.2.1 A Formula For The Inverse ........................ 85 6.2.2 Cramer's Rule ............................... 88 6.3 Exercises With Answers .............................. 90 6.4 The Mathematical Theory Of Determinants¤ . 93 6.5 The Cayley Hamilton Theorem¤ . 102 7 Rank Of A Matrix 105 7.0.1 Outcomes ..................................105 7.1 Elementary Matrices ................................105 7.2 The Row Reduced Echelon Form Of A Matrix . 111 7.3 The Rank Of A Matrix ..............................115 7.3.1 The De¯nition Of Rank . 115 7.3.2 Finding The Row And Column Space Of A Matrix . 117 7.4 Linear Independence And Bases . 118 7.4.1 Linear Independence And Dependence . 118 7.4.2 Subspaces ..................................122 7.4.3 Basis Of A Subspace ............................123 7.4.4 Extending An Independent Set To Form A Basis . 126 7.4.5 Finding The Null Space Or Kernel Of A Matrix . 127 7.4.6 Rank And Existence Of Solutions To Linear Systems . 129 7.5 Fredholm Alternative ................................129 7.5.1 Row, Column, And Determinant Rank . 131 8 Linear Transformations 135 8.0.2 Outcomes ..................................135 8.1 Linear Transformations ..............................135 8.2 Constructing The Matrix Of A Linear Transformation . 136 8.2.1 Rotations of R2 ...............................137 8.2.2 Projections .................................139 8.2.3 Matrices Which Are One To One Or Onto . 140 8.2.4 The General Solution Of A Linear System . 141 9 The LU Factorization 145 9.0.5 Outcomes ..................................145 9.1 De¯nition Of An LU factorization . 145 9.2 Finding An LU Factorization By Inspection . 145 9.3 Using Multipliers To Find An LU Factorization . 146 9.4 Solving Systems Using The LU Factorization . 147 9.5 Justi¯cation For The Multiplier Method . 148 9.6 The P LU Factorization ..............................150 9.7 The QR Factorization ...............................151 CONTENTS 5 10 Linear Programming 155 10.1 Simple Geometric Considerations . 155 10.2 The Simplex Tableau ................................156 10.3 The Simplex Algorithm ..............................160 10.3.1 Maximums .................................160 10.3.2 Minimums ..................................162 10.4 Finding A Basic Feasible Solution . 169 10.5 Duality .......................................171 11 Spectral Theory 175 11.0.1 Outcomes ..................................175 11.1 Eigenvalues And Eigenvectors Of A Matrix . 175 11.1.1 De¯nition Of Eigenvectors And Eigenvalues . 175 11.1.2 Finding Eigenvectors And Eigenvalues . 177 11.1.3 A Warning .................................179 11.1.4 Triangular Matrices ............................181 11.1.5 Defective And Nondefective Matrices . 182 11.1.6 Complex Eigenvalues ............................186 11.2 Some Applications Of Eigenvalues And Eigenvectors . 187 11.2.1 Principle Directions ............................187 11.2.2 Migration Matrices .............................188 11.3 The Estimation Of Eigenvalues . 192 11.4 Exercises With Answers ..............................193 12 Some Special Matrices 201 12.0.1 Outcomes ..................................201 12.1 Symmetric And Orthogonal Matrices . 201 12.1.1 Orthogonal Matrices ............................201 12.1.2 Symmetric And Skew Symmetric Matrices . 203 12.1.3 Diagonalizing A Symmetric Matrix . 210 12.2 Fundamental Theory And Generalizations* . 212 12.2.1 Block Multiplication Of Matrices . 212 12.2.2 Orthonormal Bases .............................215 12.2.3 Schur's Theorem¤ .............................216 12.3 Least Square Approximation . 220 12.3.1 The Least Squares Regression Line . 222 12.3.2 The Fredholm Alternative . 223 12.4 The Right Polar Factorization¤ . 223 12.5 The Singular Value Decomposition¤ . 227 13 Numerical Methods For Solving Linear Systems 231 13.0.1 Outcomes ..................................231 13.1 Iterative Methods For Linear Systems . 231 13.1.1 The Jacobi Method ............................232 13.1.2 The Gauss Seidel Method . 234 14 Numerical Methods For Solving The Eigenvalue Problem 239 14.0.3 Outcomes ..................................239 14.1 The Power Method For Eigenvalues . 239 14.2 The Shifted Inverse Power Method . 242 14.2.1 Complex Eigenvalues ............................252 14.3 The Rayleigh Quotient ...............................254 6 CONTENTS 15 Vector Spaces 259 16 Linear Transformations 265 16.1 Matrix Multiplication As A Linear Transformation . 265 16.2 L (V; W ) As A Vector Space ............................265 16.3 Eigenvalues And Eigenvectors Of Linear Transformations . 266 16.4 Block Diagonal Matrices ..............................271 16.5 The Matrix Of A Linear Transformation . 275 16.5.1 Some Geometrically De¯ned Linear Transformations . 282 16.5.2 Rotations About A Given Vector . 285 16.5.3 The Euler Angles ..............................287 A The Jordan Canonical Form* 291 B An Assortment Of Worked Exercises And Examples 299 B.1 Worked Exercises Page ?? .............................299 B.2 Worked Exercises Page ?? .............................304 B.3 Worked Exercises Page ?? .............................306 B.4 Worked Exercises Page ?? .............................309 B.5 Worked Exercises Page ?? .............................313 B.6 Worked Exercises Page ?? .............................315 B.7 Worked Exercises Page ?? .............................317 C The Fundamental Theorem Of Algebra 321 Copyright °c 2005, Introduction This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and speci¯c algorithms. At the end, the more abstract notions of vector spaces and linear transformations on vector spaces are presented. However, this is intended to be a ¯rst course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra. I have given complete proofs of all the fundamental ideas but some topics such as Markov matrices are not complete in this book but receive a plausible introduction. The book contains a complete treatment of determinants and a simple proof of the Cayley Hamilton theorem although these are optional topics. The Jordan form is presented as an appendix. I see this theorem as the beginning of more advanced topics in linear algebra and not really part of a beginning linear algebra course. There are extensions of many of the topics of this book in my on line book [9]. I have also not emphasized that linear algebra can be carried out with any ¯eld although I have done everything in terms of either the real numbers or the complex numbers. It seems to me this is a reasonable specialization for a ¯rst course in linear algebra. 7 8 INTRODUCTION Fn 2.0.1 Outcomes A. Understand the symbol, Fn in the case where F equals the real numbers, R or the complex numbers, C. B. Know how to do algebra with vectors in Fn, including vector addition and scalar multiplication. C. Understand the geometric signi¯cance of an element
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