6 CONTENTS

15.4 The Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35715.5 The QR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

15.5.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . 36115.6 MATLAB And The QR Algorithm . . . . . . . . . . . . . . . . . . . . . 362

15.6.1 The Upper Hessenberg Form . . . . . . . . . . . . . . . . . . . 36515.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

16 Vector Spaces 37316.1 Algebraic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 37316.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37516.3 Linear Independence And Bases . . . . . . . . . . . . . . . . . . . . . . 37616.4 Vector Spaces And Fields∗ . . . . . . . . . . . . . . . . . . . . . . . . . 382

16.4.1 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . 38216.4.2 Polynomials And Fields . . . . . . . . . . . . . . . . . . . . . . 38716.4.3 The Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . 39316.4.4 The Lindemannn Weierstrass Theorem And Vector Spaces . . . . 396

16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

17 Inner Product Spaces 40317.1 Basic Definitions And Examples . . . . . . . . . . . . . . . . . . . . . . 403

17.1.1 The Cauchy Schwarz Inequality And Norms . . . . . . . . . . . 40417.2 The Gram Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . 40617.3 Approximation And Least Squares . . . . . . . . . . . . . . . . . . . . . 40917.4 Orthogonal Complement . . . . . . . . . . . . . . . . . . . . . . . . . . 41317.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41317.6 The Discreet Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 41517.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

18 Linear Transformations 42518.1 Matrix Multiplication As A Linear Transformation . . . . . . . . . . . . . 42518.2 The Linear Maps as a Vector Space . . . . . . . . . . . . . . . . . . . . . 42518.3 Eigenvalues And Eigenvectors Of Linear Transformations . . . . . . . . . 42718.4 Block Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 43318.5 The Matrix Of A Linear Transformation . . . . . . . . . . . . . . . . . . 437

18.5.1 Some Geometrically Defined Linear Transformations . . . . . . 44618.5.2 Rotations About A Given Vector . . . . . . . . . . . . . . . . . 447

18.6 The Matrix Exponential, Differential Equations ∗ . . . . . . . . . . . . . . 44818.6.1 Computing A Fundamental Matrix . . . . . . . . . . . . . . . . 455

18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

A The Jordan Canonical Form* 467

B Directions For Computer Algebra Systems 475B.1 Finding Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475B.2 Finding Row Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . 475B.3 Finding PLU Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . 475B.4 Finding QR Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . 475B.5 Finding Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 475B.6 Use Of Matrix Calculator On Web . . . . . . . . . . . . . . . . . . . . . 475