4 CONTENTS

8.2 THE Row Reduced Echelon Form Of A Matrix . . . . . . . . . . . . . . 1488.3 The Rank Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.3.1 The Definition Of Rank . . . . . . . . . . . . . . . . . . . . . . 1558.3.2 Finding The Row And Column Space Of A Matrix . . . . . . . . 156

8.4 A Short Application To Chemistry . . . . . . . . . . . . . . . . . . . . . 1598.5 Linear Independence And Bases . . . . . . . . . . . . . . . . . . . . . . 160

8.5.1 Linear Independence And Dependence . . . . . . . . . . . . . . 1608.5.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.5.3 Basis Of A Subspace . . . . . . . . . . . . . . . . . . . . . . . . 1668.5.4 Extending An Independent Set To Form A Basis . . . . . . . . . 1698.5.5 Finding The Null Space Or Kernel Of A Matrix . . . . . . . . . 1708.5.6 Rank And Existence Of Solutions To Linear Systems . . . . . . 172

8.6 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.6.1 Row, Column, And Determinant Rank . . . . . . . . . . . . . . 175

8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9 Linear Transformations 1859.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.2 Constructing The Matrix Of A Linear Transformation . . . . . . . . . . . 187

9.2.1 Rotations in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1889.2.2 Rotations About A Particular Vector . . . . . . . . . . . . . . . 1909.2.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1929.2.4 Matrices Which Are One To One Or Onto . . . . . . . . . . . . 1929.2.5 The General Solution Of A Linear System . . . . . . . . . . . . 194

9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

10 A Few Factorizations 20510.1 Definition Of An LU factorization . . . . . . . . . . . . . . . . . . . . . 20510.2 Finding An LU Factorization By Inspection . . . . . . . . . . . . . . . . 20510.3 Using Multipliers To Find An LU Factorization . . . . . . . . . . . . . . 20610.4 Solving Systems Using An LU Factorization . . . . . . . . . . . . . . . . 20810.5 Justification For The Multiplier Method . . . . . . . . . . . . . . . . . . . 20810.6 The PLU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110.7 The QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21310.8 MATLAB And Factorizations . . . . . . . . . . . . . . . . . . . . . . . . 21610.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11 Linear Programming 22111.1 Simple Geometric Considerations . . . . . . . . . . . . . . . . . . . . . . 22111.2 The Simplex Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22211.3 The Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

11.3.1 Maximums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22711.3.2 Minimums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

11.4 Finding A Basic Feasible Solution . . . . . . . . . . . . . . . . . . . . . 23811.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24011.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

12 Spectral Theory 24712.1 Eigenvalues And Eigenvectors Of A Matrix . . . . . . . . . . . . . . . . 247