Linear Algebra for Computational Sciences and Engineering

Description

This book presents the main concepts of linear algebra from the viewpoint of applied scientists such as computer scientists and engineers, without compromising on mathematical rigor. Based on the idea that computational scientists and engineers need, in both research and professional life, an understanding of theoretical concepts of mathematics in order to be able to propose research advances and innovative solutions, every concept is thoroughly introduced and is accompanied by its informal interpretation. Furthermore, most of the theorems included are first rigorously proved and then shown in practice by a numerical example. When appropriate, topics are presented also by means of pseudocodes, thus highlighting the computer implementation of algebraic theory. It is structured to be accessible to everybody, from students of pure mathematics who are approaching algebra for the first time to researchers and graduate students in applied sciences who need a theoretical manual of algebra to successfully perform their research. Most importantly, this book is designed to be ideal for both theoretical and practical minds and to offer to both alternative and complementary perspectives to study and understand linear algebra. Foreword Preface to the Second Edition Preface to the First Edition Contents About the Author Part I Foundations of Linear Algebra 1 Basic Mathematical Thinking 1.1 Introduction 1.2 Axiomatic System 1.3 Basic Definitions in Set Theory 1.3.1 Order and Equivalence 1.4 Functions 1.5 Number Sets 1.6 A Preliminary Introduction to Algebraic Structures 2 Matrices 2.1 Numeric Vectors 2.2 Basic Definitions About Matrices 2.3 Matrix Operations 2.4 Determinant of a Matrix 2.4.1 Linear Dependence of Row and Column Vectors of a Matrix 2.4.2 Properties of the Determinant 2.4.3 Submatrices, Cofactors and Adjugate Matrices 2.4.4 Laplace Theorems on Determinants 2.5 Invertible Matrices 2.6 Orthogonal Matrices 2.7 Rank of a Matrix 3 Systems of Linear Equations 3.1 Solution of a System of Linear Equations 3.2 Homogeneous Systems of Linear Equations 3.3 Direct Methods 3.3.1 Gaussian Elimination 3.3.1.1 Row Vector Notation for Gaussian Elimination 3.3.2 Pivoting Strategies and Computational Cost 3.3.3 LU Factorization 3.3.4 Equivalence of Gaussian Elimination and LU Factorization 3.4 Iterative Methods 3.4.1 Jacobi's Method 3.4.2 Gauss-Seidel's Method 3.4.3 The Method of Successive Over Relaxation 3.4.4 Numerical Comparison Among the Methods and Convergence Conditions 4 Geometric Vectors 4.1 Basic Concepts 4.2 Linear Dependence and Linear Independence 4.3 Matrices of Vectors 4.4 Bases of Vectors 4.5 Products of Vectors 5 Complex Numbers and Polynomials 5.1 Complex Numbers 5.2 Complex Vectors, Matrices and Systems of Linear Equation 5.3 Complex Polynomials 5.3.1 Operations of Polynomials 5.3.2 Roots of Polynomials 5.3.2.1 How to Determine the Roots of a Polynomial 5.4 Partial Fractions 6 An Introduction to Geometric Algebra and Conics 6.1 Basic Concepts: Lines in the Plane 6.1.1 Equations of the Line 6.1.2 Intersecting Lines 6.1.3 Families of Straight Lines 6.2 An Intuitive Introduction to the Conics 6.3 Analytical Representation of a Conic 6.4 Simplified Representation of Conics 6.4.1 Simplified Representation of Degenerate Conics 6.4.2 Simplified Representation of Non-degenerate Conics 6.5 Matrix Representation of a Conic 6.5.1 Intersection with a Line 6.5.2 Line Tangent to a Conic 6.5.3 Degenerate and Non-degenerate Conics: A Conic as a Matrix 6.5.4 Classification of a Conic: Asymptotic Directions of a Conic 6.5.5 Diameters, Centres, Asymptotes, and Axes of Conics 6.5.6 Canonic Form of a Conic Part II Elements of Linear Algebra 7 An Overview on Algebraic Structures 7.1 Basic Concepts 7.2 Semigroups and Monoids 7.3 Groups and Subgroups 7.3.1 Cosets 7.3.2 Equivalence and Congruence Relation 7.3.3 Lagrange's Theorem 7.4 Rings 7.4.1 Cancellation Law for Rings 7.4.2 Fields 7.5 Homomorphisms and Isomorphisms 8 Vector Spaces 8.1 Basic Concepts 8.2 Vector Subspaces 8.3 Linear Dependence in n Dimensions 8.4 Linear Span 8.5 Basis and Dimension of a Vector Space 8.6 Row and Column Spaces 9 An Introduction to Inner Product Spaces: Euclidean Spaces 9.1 Basic Concepts: Inner Products 9.2 Euclidean Spaces 9.3 Euclidean Spaces in Two Dimensions 9.4 Gram-Schmidt Orthonormalization 10 Linear Mappings 10.1 Introductory Concepts 10.2 Linear Mappings and Vector Spaces 10.3 Endomorphisms and Kernel 10.4 Rank and Nullity of Linear Mappings 10.4.1 Matrix Representation of a Linear Mapping 10.4.2 A Linear Mapping as a Matrix: A Summarizing Scheme 10.4.3 Invertible Mappings 10.4.4 Similar Matrices 10.4.5 Geometric Mappings 10.5 Eigenvalues, Eigenvectors, and Eigenspaces 10.5.1 Method for Determining Eigenvalues and Eigenvectors 10.6 Matrix Diagonalization 10.6.1 Diagonalization of a Symmetric Mapping 10.7 Power Method 11 An Introduction to Computational Complexity 11.1 Complexity of Algorithms and Big-O Notation 11.2 P, NP, NP-Hard, NP-Complete Problems 11.3 Representing Information 11.3.1 Huffman Coding 11.3.2 Polish and Reverse Polish Notation 12 Graph Theory 12.1 Motivation and Basic Concepts 12.2 Eulerian and Hamiltonian Graphs 12.3 Bipartite Graphs 12.4 Planar Graphs 12.4.1 Trees and Cotrees 12.4.1.1 Analogy Between Graphs and Vector Spaces 12.4.2 Euler's Formula 12.5 Graph Matrices 12.5.1 Adjacency Matrices 12.5.2 Incidence Matrices 12.5.3 Cycle Matrices 12.5.4 Cut-Set Matrices 12.5.5 Relation Among Fundamental Matrices 12.5.5.1 Graph Matrices and Vector Spaces 12.6 Graph Isomorphisms and Automorphisms 12.7 Some Applications of Graph Theory 12.7.1 The Social Network Problem 12.7.2 The Four Colour Problem 12.7.3 Travelling Salesman Problem 12.7.4 The Chinese Postman Problem 12.7.5 Applications to Sociology or to the Spread of Epidemics 13 Applied Linear Algebra: Electrical Networks 13.1 Basic Concepts 13.2 Bi-poles 13.2.1 Passive Bi-poles 13.2.2 Active Bi-poles 13.3 Electrical Networks and Circuits 13.3.1 Bi-poles in Series and Parallel 13.3.2 Kirchoff's Laws 13.3.3 Phasorial Representation of Electrical Quantities 13.3.4 Impedance 13.4 Solving Electrical Networks 13.5 Remark Appendix A A Non-linear Algebra: An Introduction to Boolean Algebra A.1 Basic Logical Gates A.2 Properties of Boolean Algebra A.3 Boolean Algebra in Algebraic Structures A.4 Composed Boolean Gates A.5 Crisp and Fuzzy sets Appendix B Proofs of Theorems That Require Further Knowledge of Mathematics Solutions to the Exercises References Index