Linear Algebra with Python. Theory and Applications

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