A First Course in Applied Mathematics

CONTENTS

Preface

1 Basics of Linear Algebra

1.1 Notation and Terminology

1.2 Vector and Matrix Norms

1.3 Dot Product and Orthogonality

1.4 Special Matrices

1.4.1 Diagonal and triangular matrices

1.4.2 Hessenberg matrices

1.4.3 Nonsingular and inverse matrices

1.4.4 Symmetric and positive definite matrices

1.4.5 Matrix exponential

1.4.6 Permutation matrices

1.4.7 Orthogonal matrices

1.5 Vector Spaces

1.6 Linear Independence and Basis

1.7 Orthogonalization and Direct Sums

1.8 Column Space, Row Space, and Null Space

1.8.1 Linear transformations

1.9 Orthogonal Projections

1.10 Eigenvalues and Eigenvectors

1.11 Similarity

1.12 Bezier Curves and Postscript Fonts

1.12.1 Properties of Bezier curves

1.12.2 Composite Bezier curves

1.13 Final Remarks and Further Reading

Exercises

2 Ranking Web Pages

2.1 The Power Method

2.2 Stochastic, Irreducible, and Primitive Matrices

2.3 Google's PageRank Algorithm

2.3.1 The personalization vector

2.3.2 Speed of convergence and sparsity

2.3.3 Power method and reordering

2.4 Alternatives to the Power Method

2.4.1 Linear system formulation

2.4.2 Iterative aggregation/disaggregation (IAD)

2.4.3 IAD and linear systems

2.5 Final Remarks and Further Reading

Exercises

3 Matrix Factorizations

3.1 LU Factorization

3.1.1 The complex case

3.1.2 Solving several systems

3.1.3 The PA = LU factorization

3.2 QR Factorization

3.2.1 QR and Gram–Schmidt

3.2.2 The complex case

3.2.3 QR and similarity

3.2.4 The QR algorithm

3.2.5 QR and LU

3.3 Singular Value Decomposition ...