Paul Dawkins

Linear Algebra

Table of Contents

Preface............................................................................................................................................. ii Outline............................................................................................................................................ iii Systems of Equations and Matrices.............................................................................................. 1 Introduction ................................................................................................................................................ 1 Systems of Equations ................................................................................................................................. 3 Solving Systems of Equations .................................................................................................................. 15 Matrices .................................................................................................................................................... 27 Matrix Arithmetic & Operations .............................................................................................................. 33 Properties of Matrix Arithmetic and the Transpose ................................................................................. 45 Inverse Matrices and Elementary Matrices .............................................................................................. 50 Finding Inverse Matrices .......................................................................................................................... 59 Special Matrices ....................................................................................................................................... 68 LU-Decomposition ................................................................................................................................... 75 Systems Revisited .................................................................................................................................... 81

Determinants ................................................................................................................................ 90 Introduction .............................................................................................................................................. 90 The Determinant Function ....................................................................................................................... 91 Properties of Determinants ..................................................................................................................... 100 The Method of Cofactors ....................................................................................................................... 107 Using Row Reduction To Compute Determinants ................................................................................. 115 Cramer’s Rule ........................................................................................................................................ 122

Euclidean n-Space ...................................................................................................................... 125 Introduction ............................................................................................................................................ 125 Vectors ................................................................................................................................................... 126 Dot Product & Cross Product ................................................................................................................. 140 Euclidean n-Space .................................................................................................................................. 154 Linear Transformations...