Applied Linear Algebra: Second Edition

This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems. No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author's text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here. -- Back cover. Read more... Abstract: This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems. No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author's text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here. -- Back cover Preface Syllabi and Prerequisites Survey of Topics Course Outlines Comments on Individual Chapters Changes from the First Edition Exercises and Software Conventions and Notations History and Biography Some Final Remarks Acknowledgments Table of Contents Chapter 1: Linear Algebraic Systems 1.1 Solution of Linear Systems 1.2 Matrices and Vectors Matrix Arithmetic 1.3 Gaussian Elimination—Regular Case Elementary Matrices The LU Factorization Forward and Back Substitution 1.4 Pivoting and Permutations Permutations and Permutation Matrices The Permuted LU Factorization 1.5 Matrix Inverses Gauss–Jordan Elimination Solving Linear Systems with the Inverse The LDV Factorization 1.6 Transposes and Symmetric Matrices Factorization of Symmetric Matrices 1.7 Practical Linear Algebra Tridiagonal Matrices Pivoting Strategies 1.8 General Linear Systems Homogeneous Systems 1.9 Determinants Chapter 2: Vector Spaces and Bases 2.1 Real Vector Spaces 2.2 Subspaces 2.3 Span and Linear Independence Linear Independence and Dependence 2.4 Basis and Dimension 2.5 The Fundamental Matrix Subspaces Kernel and Image The Superposition Principle Adjoint Systems, Cokernel, and Coimage The Fundamental Theorem of Linear Algebra 2.6 Graphs and Digraphs Chapter 3: Inner Products and Norms 3.1 Inner Products Inner Products on Function Spaces 3.2 Inequalities The Cauchy–Schwarz Inequality Orthogonal Vectors The Triangle Inequality 3.3 Norms Unit Vectors Equivalence of Norms Matrix Norms 3.4 Positive Definite Matrices Gram Matrices 3.5 Completing the Square The Cholesky Factorization 3.6 Complex Vector Spaces Complex Numbers Complex Vector Spaces and Inner Products Chapter 4: Orthogonality 4.1 Orthogonal and Orthonormal Bases Computations in Orthogonal Bases 4.2 The Gram–Schmidt Process Modifications of the Gram–Schmidt Process 4.3 Orthogonal Matrices The QR Factorization Ill-Conditioned Systems and Householder’s Method 4.4 Orthogonal Projections and Orthogonal Subspaces Orthogonal Projection Orthogonal Subspaces Orthogonality of the Fundamental Matrix Subspaces and the Fredholm Alternative 4.5 Orthogonal Polynomials The Legendre Polynomials Other Systems of Orthogonal Polynomials Chapter 5: Minimization and Least Squares 5.1 Minimization Problems Equilibrium Mechanics Solution of Equations The Closest Point 5.2 Minimization of Quadratic Functions 5.3 The Closest Point 5.4 Least Squares 5.5 Data Fitting and Interpolation Polynomial Approximation and Interpolation Approximation and Interpolation by General Functions Least Squares Approximation in Function Spaces Orthogonal Polynomials and Least Squares Splines 5.6 Discrete Fourier Analysis and the Fast Fourier Transform Compression and Denoising The Fast Fourier Transform Chapter 6: Equilibrium 6.1 Springs and Masses Positive Definiteness and the Minimization Principle 6.2 Electrical Networks Batteries, Power, and the Electrical–Mechanical Correspondence 6.3 Structures Chapter 7: Linearity 7.1 Linear Functions Linear Operators The Space of Linear Functions Dual Spaces Composition Inverses 7.2 Linear Transformations Change of Basis 7.3 Affine Transformations and Isometries Isometry 7.4 Linear Systems The Superposition Principle Inhomogeneous Systems Superposition Principles for Inhomogeneous Systems Complex Solutions to Real Systems 7.5 Adjoints, Positive Definite Operators, and Minimization Principles Self-Adjoint and Positive Definite Linear Functions Minimization Chapter 8: Eigenvalues and Singular Values 8.1 Linear Dynamical Systems Scalar Ordinary Differential Equations First Order Dynamical Systems 8.2 Eigenvalues and Eigenvectors Basic Properties of Eigenvalues The Gershgorin Circle Theorem 8.3 Eigenvector Bases Diagonalization 8.4 Invariant Subspaces 8.5 Eigenvalues of Symmetric Matrices The Spectral Theorem Optimization Principles for Eigenvalues of Symmetric Matrices 8.6 Incomplete Matrices The Schur Decomposition The Jordan Canonical Form 8.7 Singular Values The Pseudoinverse The Euclidean Matrix Norm Condition Number and Rank Spectral Graph Theory 8.8 Principal Component Analysis Variance and Covariance The Principal Components Chapter 9: Iteration 9.1 Linear Iterative Systems Scalar Systems Powers of Matrices Diagonalization and Iteration 9.2 Stability Spectral Radius Fixed Points Matrix Norms and Convergence 9.3 Markov Processes 9.4 Iterative Solution of Linear Algebraic Systems The Jacobi Method The Gauss–Seidel Method Successive Over-Relaxation 9.5 Numerical Computation of Eigenvalues The Power Method The QR Algorithm Tridiagonalization 9.6 Krylov Subspace Methods Krylov Subspaces Arnoldi Iteration The Full Orthogonalization Method The Conjugate Gradient Method The Generalized Minimal Residual Method 9.7 Wavelets The Haar Wavelets Modern Wavelets Solving the Dilation Equation Chapter 10: Dynamics 10.1 Basic Solution Techniques The Phase Plane Existence and Uniqueness Complete Systems The General Case 10.2 Stability of Linear Systems 10.3 Two-Dimensional Systems Distinct Real Eigenvalues Complex Conjugate Eigenvalues Incomplete Double Real Eigenvalue Complete Double Real Eigenvalue 10.4 Matrix Exponentials Applications in Geometry Invariant Subspaces and Linear Dynamical Systems Inhomogeneous Linear Systems 10.5 Dynamics of Structures Stable Structures Unstable Structures Systems with Differing Masses Friction and Damping 10.6 Forcing and Resonance Electrical Circuits Forcing and Resonance in Systems References Symbol Index Subject Index Main subject categories: • Linear algebra • Ordinary differential equationsThis textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems.No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author’s text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here. "This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems. No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author’s text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here". -- Prové de l'editor