A First Course in Applied Mathematics

Explore real-world applications of selected mathematical theory, concepts, and methods

Exploring related methods that can be utilized in various fields of practice from science and engineering to business, A First Course in Applied Mathematics details how applied mathematics involves predictions, interpretations, analysis, and mathematical modeling to solve real-world problems.

Written at a level that is accessible to readers from a wide range of scientific and engineering fields, the book masterfully blends standard topics with modern areas of application and provides the needed foundation for transitioning to more advanced subjects. The author utilizes MATLAB® to showcase the presented theory and illustrate interesting real-world applications to Google's web page ranking algorithm, image compression, cryptography, chaos, and waste management systems. Additional topics covered include:

• Linear algebra

• Ranking web pages

• Matrix factorizations

• Least squares

• Image compression

• Ordinary differential equations

• Dynamical systems

• Mathematical models

Throughout the book, theoretical and applications-oriented problems and exercises allow readers to test their comprehension of the presented material. An accompanying website features related MATLAB® code and additional resources.

A First Course in Applied Mathematics is an ideal book for mathematics, computer science, and engineering courses at the upper-undergraduate level. The book also serves as a valuable reference for practitioners working with mathematical modeling, computational methods, and the applications of mathematics in their everyday work.

Preface xiii

1 Basics of Linear Algebra 1

1.1 Notation and Terminology 1 

1.2 Vector and Matrix Norms 4

1.3 Dot Product and Orthogonality 8 

1.4 Special Matrices 9

1.5 Vector Spaces 21

1.6 Linear Independence and Basis 24

1.7 Orthogonalization and Direct Sums 30

1.8 Column Space, Row Space and Null Space 34

1.9 Orthogonal Projections 43

1.10 Eigenvalues and Eigenvectors 47

1.11 Similarity 56

1.12 Bezier Curves Postscript Fonts 59

1.13 Final Remarks and Further Reading 68

Exercises 69

2 Ranking Web Pages 79

2.1 The Power Method 80

2.2 Stochastic, Irreducible and Primitive Matrices 84

2.3 Google’s PageRank Algorithm 92

2.4 Alternatives to Power Method 106

2.5 Final Remarks and Further Reading 120

Exercises 121

3 Matrix Factorizations 131 

3.1 LU Factorization 132

3.2 QR Factorization 142

3.3 Singular Value Decomposition (SVD) 155

3.4 Schur Factorization 166

3.5 Information Retrieval 186

3.6 Partition of Simple Substitution Cryptograms 194

3.7 Final Remarks and Further Reading 203

Exercises 205

4 Least Squares 215

4.1 Projections and Normal Equations 215

4.2 Least Squares and QR Factorization 224

4.3 Lagrange Multipliers 228

4.4 Final Remarks and Further Reading 231

Exercises 231

5 Image Compression 235

5.1 Compressing with Discrete Cosine Transform 236

5.2 Huffman Coding 260

5.3 Compression with SVD 267

5.4 Final Remarks and Further Reading 269

Exercises 271

6 Ordinary Differential Equations 277

6.1 One-Dimensional Differential Equations 278

6.2 Linear Systems of Differential Equations 307

6.3 Solutions via Eigenvalues and Eigenvectors 307

6.4 Fundamentals Matrix Solution 312

6.5 Final Remarks and Further Reading 316

Exercises 316

7 Dynamical Systems 325

7.1 Linear Dynamical Systems 326

7.2 Nonlinear Dynamical Systems 340

7.3 Predator-Prey Models with Harvesting 374

7.4 Final Remarks and Further Reading 385

Exercises 385

8 Mathematical Models 395

8.1 Optimization of a Waste Management System 396

8.2 Grouping Problem in Networks 404 

8.3 American Cutaneous Leishmaniasis 410

8.4 Variable Population Interactions 420

References 431

Index 435

JORGE REBAZA, PHD, is Associate Professor in the Department of Mathematics at Missouri State University. Dr. Rebaza has published numerous journal articles in his areas of research interest, which include numerical analysis, dynamical systems, matrix computations, and applied mathematics.