MATH 545 Elementary Linear Algebra Kuttler

CONTENTS 1 Some Prerequisite Topics 1 1.1 Sets And Set Notation ................................... 1 1.2 Well Ordering And Induction ............................... 2 1.3 The Complex Numbers ................................... 4 1.4 Polar Form Of Complex Numbers ............................. 6 1.5 Roots Of Complex Numbers ................................ 6 1.6 The Quadratic Formula .................................. 8 1.7 The Complex Exponential ................................. 9 1.8 Exercises .......................................... 9 2 Fn 13 2.1 Algebra in Fn ........................................ 14 2.2 Geometric Meaning Of Vectors .............................. 15 2.3 Geometric Meaning Of Vector Addition ......................... 16 2.4 Distance Between Points In Rn Length Of A Vector .................. 17 2.5 Geometric Meaning Of Scalar Multiplication ...................... 20 2.6 Parametric Lines ...................................... 21 2.7 Exercises .......................................... 22 2.8 Vectors And Physics .................................... 22 2.9 Exercises .......................................... 24 3 Vector Products 27 3.1 The Dot Product ...................................... 27 3.2 The Geometric Significance Of The Dot Product .................... 29 3.2.1 The Angle Between Two Vectors ......................... 29 3.2.2 Work And Projections ............................... 30 3.2.3 The Inner Product And Distance In Cn ..................... 32 3.3 Exercises .......................................... 35 3.4 The Cross Product ..................................... 36 3.4.1 The Distributive Law For The Cross Product .................. 39 3.4.2 The Box Product .................................. 40 3.4.3 A Proof Of The Distributive Law ......................... 41 3.5 The Vector Identity Machine ............................... 42 3.6 Exercises .......................................... 43 4 Systems Of Equations 45 4.1 Systems Of Equations, Geometry ............................. 45 4.2 Systems Of Equations, Algebraic Procedures ...................... 47 4.2.1 Elementary Operations .............................. 47 4.2.2 Gauss Elimination ................................. 49 4.2.3 Balancing Chemical Reactions .......................... 57 4.2.4 Dimensionless Variables§ ............................. 59 4.3 Exercises .......................................... 61 3 4 CONTENTS 5 Matrices 67 5.1 Matrix Arithmetic ..................................... 67 5.1.1 Addition And Scalar Multiplication Of Matrices ................ 67 5.1.2 Multiplication Of Matrices ............................ 69 5.1.3 The ijth Entry Of A Product ........................... 72 5.1.4 Properties Of Matrix Multiplication ....................... 74 5.1.5 The Transpose ................................... 75 5.1.6 The Identity And Inverses ............................. 76 5.1.7 Finding The Inverse Of A Matrix ......................... 77 5.2 Exercises .......................................... 81 6 Determinants 87 6.1 Basic Techniques And Properties ............................. 87 6.1.1 Cofactors And 2 £ 2 Determinants ........................ 87 6.1.2 The Determinant Of A Triangular Matrix .................... 90 6.1.3 Properties Of Determinants ............................ 91 6.1.4 Finding Determinants Using Row Operations .................. 92 6.2 Applications ......................................... 94 6.2.1 A Formula For The Inverse ............................ 94 6.2.2 Cramer’s Rule ................................... 97 6.3 Exercises .......................................... 99 7 The Mathematical Theory Of Determinants§ 105 7.1 The Function sgnn ..................................... 105 7.2 The Determinant ...................................... 107 7.2.1 The Definition ................................... 107 7.2.2 Permuting Rows Or Columns ........................... 107 7.2.3 A Symmetric Definition .............................. 108 7.2.4 The Alternating Property Of The Determinant ................. 109 7.2.5 Linear Combinations And Determinants ..................... 110 7.2.6 The Determinant Of A Product .......................... 110 7.2.7 Cofactor Expansions ................................ 111 7.2.8 Formula For The Inverse .............................. 112 7.2.9 Cramer’s Rule ................................... 113 7.2.10 Upper Triangular Matrices ............................ 114 7.3 The Cayley Hamilton Theorem§ ............................. 114 8 Rank Of A Matrix 117 8.1 Elementary Matrices .................................... 117 8.2 THE Row Reduced Echelon Form Of A Matrix ..................... 122 8.3 The Rank Of A Matrix .................................. 127 8.3.1 The Definition Of Rank .............................. 127 8.3.2 Finding The Row And Column Space Of A Matrix ............... 128 8.4 A Short Application To Chemistry ............................ 130 8.5 Linear Independence And Bases .............................. 131 8.5.1 Linear Independence And Dependence ...................... 131 8.5.2 Subspaces ...................................... 134 8.5.3 Basis Of A Subspace ................................ 136 8.5.4 Extending An Independent Set To Form A Basis ................ 139 8.5.5 Finding The Null Space Or Kernel Of A Matrix ................ 139 8.5.6 Rank And Existence Of Solutions To Linear Systems .............. 141 8.6 Fredholm Alternative .................................... 142 8.6.1 Row, Column, And Determinant Rank ...................... 143 8.7 Exercises .......................................... 146