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Linear Algebra with Python (2023) [Tsukada et al] [9789819929511] Preface Contents 1 Mathematics and Python 1.1 Propositional Logic 1.2 Numbers 1.3 Sets 1.4 Ordered Pairs and Tuples 1.5 Mappings and Functions 1.6 Classes and Objects in Python 1.7 Lists, Arrays and Matrices 1.8 Preparation of Image Data 1.8.1 Binarization of Image Data with PIL and NumPy 1.8.2 GUI for Creating Complex-Valued Data of Handwritten Characters 1.8.3 Data of Handwritten Letters with Grayscale 2 Linear Spaces and Linear Mappings 2.1 Linear Spaces 2.2 Subspaces 2.3 Linear Mappings 2.4 Application: Visualizing Sounds 3 Basis and Dimension 3.1 Finite-Dimensional Linear Spaces 3.2 Linear Dependence and Linear Independence 3.3 Basis and Representation 3.4 Dimension and Rank 3.5 Direct Sums 3.6 Remarks on Dimension 4 Matrices 4.1 Matrix Operations 4.2 Matrices and Linear Mappings 4.3 Composition of Linear Mappings and Product of Matrices 4.4 Inverse Matrix, Basis Change, and Similarity of Matrices 4.5 Adjoint Matrix 4.6 Measuring Matrix Computation Time 5 Elementary Operations and Matrix Invariants 5.1 Elementary Matrices and Operations 5.2 Rank 5.3 Determinant 5.4 Trace 5.5 Systems of Linear Equations 5.6 Inverse Matrix 6 Inner Product and Fourier Expansion 6.1 Norm and Inner Product 6.2 Orthonormal Systems and Fourier Transform 6.3 Function Spaces 6.4 Least Squares, Trigonometric Series, and Fourier Series 6.5 Orthogonal Function Systems 6.6 Convergence of Vector Sequences 6.7 Fourier Analysis 7 Eigenvalues and Eigenvectors 7.1 Unitary Matrices and Hermitian Matrices 7.2 Eigenvalues 7.3 Diagonalization 7.4 Matrix Norm and Matrix Functions 8 Jordan Normal Form and Spectrum 8.1 Direct Sum Decomposition 8.2 Jordan Normal Form 8.3 Jordan Decomposition and Matrix Power 8.4 Spectrum of a Matrix 8.5 Perron–Frobenius Theorem 9 Dynamical Systems 9.1 Differentiation of Vector-(Matrix-) Valued Functions 9.2 Newton's Equation of Motion 9.3 Linear Differential Equations 9.4 Stationary States of Markov Chain 9.5 Markov Random Fields 9.6 One-Parameter Semigroups 10 Applications and Development of Linear Algebra 10.1 Linear Equations and Least Squares 10.2 Generalized Inverse and Singular Value Decomposition 10.3 Tensor Products 10.4 Tensor Product Representation of Vector-Valued Random Variables 10.5 Principal Component Analysis and KL Expansion 10.6 Estimation of Random Variables by Linear Regression Models 10.7 Kalman Filter Appendix A.1 Python Environment Used in This Book A.1.1 Windows A.1.2 macOS A.1.3 Raspberry Pi OS A.2 Launching Python A.3 Using Jupyter Notebook A.4 Using Libraries A.5 Python Syntax A.6 Other Tools (Supplementary) Afterword and Bibliography Symbol Index Python Index Index This textbook is for those who want to learn linear algebra from the basics. After a brief mathematical introduction, it provides the standard curriculum of linear algebra based on an abstract linear space. It covers, among other aspects: linear mappings and their matrix representations, basis, and dimension; matrix invariants, inner products, and norms; eigenvalues and eigenvectors; and Jordan normal forms. Detailed and self-contained proofs as well as descriptions are given for all theorems, formulas, and algorithms. A unified overview of linear structures is presented by developing linear algebra from the perspective of functional analysis. Advanced topics such as function space are taken up, along with Fourier analysis, the Perron–Frobenius theorem, linear differential equations, the state transition matrix and the generalized inverse matrix, singular value decomposition, tensor products, and linear regression models. These all provide a bridge to more specialized theories based on linear algebra in mathematics, physics, engineering, economics, and social sciences. Python is used throughout the book to explain linear algebra. Learning with Python interactively, readers will naturally become accustomed to Python coding. By using Python's libraries NumPy, Matplotlib, VPython, and SymPy, readers can easily perform large-scale matrix calculations, visualization of calculation results, and symbolic computations. All the codes in this book can be executed on both Windows and macOS and also on Raspberry Pi.