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2023 | Book

Mathematics and Python Read chapter Read first chapter

Linear Algebra with Python

Theory and Applications

Authors: Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi

Publisher: Springer Nature Singapore

Book Series : Springer Undergraduate Texts in Mathematics and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Mathematics and Python
Abstract
In this chapter, we survey the notions of propositions, real and complex numbers, sets, and mappings (functions) that are the basics of mathematics needed to study linear algebra. We will also learn how these concepts are expressed in Python. Let us learn mathematics in a practical way by using Python.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 2. Linear Spaces and Linear Mappings
Abstract
Based on the mathematical preparations in the previous chapter, we will learn about a linear space, the stage on which linear algebra is played, and a linear mapping which plays a leading part in linear algebra.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 3. Basis and Dimension
Abstract
In this chapter, we learn the notions of “subspace generation” and “linear independence”, which play essential roles in linear algebra. These yield the notions of “basis” and “dimension”, and they turn out to be useful tools to analyze a linear space. Some important theorems concerning them will appear. The notion of linear independence is very important, and the readers are encouraged to understand its meaning to proceed to the next step.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 4. Matrices
Abstract
In this chapter, we will learn the matrix representation of a linear mapping, and matrix operations defined naturally through representation.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 5. Elementary Operations and Matrix Invariants
Abstract
In the previous chapter, we learned the matrix representation \(\boldsymbol{A}\) of a linear mapping \(\boldsymbol{f}:V\rightarrow W\).
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 6. Inner Product and Fourier Expansion
Abstract
In this chapter, we consider a scalar-valued binary operation on a linear space called an inner product. It leads to the concepts of the length of a vector and the orthogonality between vectors, which give to a linear space the structure of Euclidean geometry. Also, we learn the meaning of orthogonality between functions in a function space.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 7. Eigenvalues and Eigenvectors
Abstract
This chapter deals with the matrix eigenvalue problem, another major theme in linear algebra as important as the theory of linear equations. This problem is based on the fundamental theorem of algebra, which states that any polynomial with complex coefficients has a complex root. We will give a short proof of it. Though the proof needs some advanced knowledge of analysis, the reader will be able to grasp it with a little effort.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 8. Jordan Normal Form and Spectrum
Abstract
We have seen that a necessary and sufficient condition for a matrix to be diagonalizable is that a set of its eigenvectors forms a basis of the underlying linear space. In the first half of this chapter, we study the Jordan normal form and the Jordan decomposition which generalize the above fact for arbitrary matrices not necessarily diagonalizable. We explain how to compute them in Python for large matrices which may be hard and cumbersome using only paper and pencil. We also make a program which generates classroom or examination problems.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 9. Dynamical Systems
Abstract
Dynamical systems are mathematical models concerning variables that change depending on time. In this chapter, we consider a system represented by a linear differential equation in which a variable changes deterministically over continuously changing time, and a Markov chain in which a variable changes stochastically with discretely changing time. In studying these systems, the theory of linear algebra, which we have learned so far, will play a major role.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Chapter 10. Applications and Development of Linear Algebra
Abstract
In this chapter, we integrate what we have learned so far into the theory of singular value decompositions and generalized inverse matrices, and discuss related topics. These theories can be considered as generalizations of Fourier analysis for orthonormal systems and as the theory of linear equations for non-regular matrices. They are closely linked to the eigenvalue problem, and because only Hermitian matrices (or real symmetric matrices) appear, we can handle them relatively easily both theoretically and computationally.
Makoto Tsukada, Yuji Kobayashi, Hiroshi Kaneko, Sin-Ei Takahasi, Kiyoshi Shirayanagi, Masato Noguchi
Backmatter
Metadata
Title
Linear Algebra with Python
Authors
Makoto Tsukada
Yuji Kobayashi
Hiroshi Kaneko
Sin-Ei Takahasi
Kiyoshi Shirayanagi
Masato Noguchi
Copyright Year
2023
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-9929-51-1
Print ISBN
978-981-9929-50-4
DOI
https://doi.org/10.1007/978-981-99-2951-1

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