Linear Algebra (eBook)

A Minimal Polynomial Approach to Eigen Theory

Fernando Barrera-Mora (Autor)

eBook Download: EPUB

2023
312 Seiten
De Gruyter (Verlag)
978-3-11-113614-1 (ISBN)

Infos

There are numerous linear algebra textbooks available on the market. Yet, there are few that approach the notion of eigenvectors and eigenvalues across an operator's minimum polynomial.

In this book, we take that approach. This book provides a thorough introduction to the fundamental concepts of linear algebra. The material is divided into two sections: Part I covers fundamental concepts in linear algebra, whereas Part II covers the theory of determinants, the theory of eigenvalues and eigenvectors, and fundamental results on Euclidean vector spaces. We highlight that:

Consider hypothetical manufacturing models as a starting point for studying linear equations. There are two novel ideas in the book: the use of a production model to motivate the concept of matrix product and the use of an operator's minimal polynomial to describe the theory of eigenvalues and eigenvectors.

Several examples incorporate the use of SageMath., allowing the reader to focus on conceptual comprehension rather than formulas.

Fernando Barrera-Mora is a Professor of Mathematics at Autonomous University of Hidalgo State (UAEH) Mexico. He was head of the Mathematics Academics Area at UAEH from 2010 to 2017. He obtained a PhD in Mathematics at the University of Arizona in 1989. He has been a member of The Sistema Nacional de Investigadores (National System of Researchers) in Mexico for over 20 years. His research interests include Number Theory and Math Education. His most recent research and publications have focused on using instructional technology to enhance problem solving strategies. When he's not working Fernando enjoys early morning hikes and occasionally plays chess for fun.

Textauszug

This work is dedicated to the memory of my father, Antonio Barrera Pelcastre (1903–2005), from whom I learned that work makes men respectful and that a deep friendship between them is based on esprit de corps work.

Foreword

One of the most popular topics in mathematics, where many textbooks have been written, is linear algebra. This is not a coincidence at all. Linear algebra appears naturally in practically all disciplines, both within mathematics and other sciences, including areas of social sciences and humanities, having significant presence in engineering and especially in physics.

From our first studies, let us say at the level of high school, linear algebra is utilized, even if such a name is not used, for solving linear equations.

Due to the above, when a new textbook related to linear algebra appears in the market, one wonders what this new textbook can contribute with, that has not been presented and studied ad nauseam in the innumerable texts that already exist.

Fernando Barrera Mora’s book is dedicated to the linear algebra of a first course of a college major in mathematics. The first thing that I would like to point out is that the book privileges starting with concrete problems that are presented to us in our daily life as in economy, in productive companies, etc. These concrete problems are the starting point to elaborate and combine the ingredients to help students visualize and relate these components, when they are raised to more general situations.

Also, these specific problems that are studied serve to establish both the methods and the necessary theory either to solve them, to study them, or to place them in a more general context.

A valuable point of view to highlight in this work is the approach to what we could call the “eigentheory,” that is, the theory that deals with “eigenvalues” (proper values) and “eigenvectors” (proper vectors). This is the heart of the book.

The most common approach to address the solution and study of eigenvalues and eigenvectors is the study of the characteristic matrix, that is, one needs to find the values for which this matrix is singular, which brings us immediately to the calculation of its determinant and, therefore, to the characteristic polynomial.

Although the analysis of the determinants is indispensable in the study of linear algebra, presenting a comprehensive study of its basic properties is a laborious or unclear problem, depending on the approach we select.

In this work, a different approach is taken. The main focus of interest is on the minimal polynomial rather than in the characteristic one. The emphasis is placed on properties inherent to the matrix that gives rise to the problem under study. More precisely, the operator associated to the matrix is studied with respect to a basis appropriately selected. In addition to the natural advantage of having an intrinsic study of the operator, the presentation given here makes it unnecessary to invoke determinants. In any case, the determinants are introduced as areas and volumes of parallelepipeds in two and three dimensions, respectively.

There are several other novelties that differentiate this text from others. For instance, in this work linear algebra is made to interact with analytic geometry; subspaces are introduced and worked with, without even having formally defined what a vector space is; there are several novel or less known proofs, such as the existence of the adjoint operator or that any linearly independent system has cardinality less than or equal to the cardinality of a set of generators; the necessary theoretical base is constructed from a two-dimensional space, it is passed to a three-dimensional space, and finally to any finite-dimensional space; an algorithm is presented for the computation of the minimum polynomial of a matrix.

Other aspects worth mentioning are the way in which the product of matrices is motivated, which is derived from a concrete example about production, and that there are included several exercises, either original or uncommon in other textbooks.

The book may very well be divided into two parts. The first consisting of the first five chapters that have a more basic nature than the other four. The first part introduces the student to the study of linear algebra and the second part is a higher level study.

A final point that needs to be emphasized is, as mentioned at the beginning, that linear algebra is of great importance in all science and engineering curricula and even in other disciplines. This importance is found in the mind of the author throughout this book, which can be perceived by the conception of linear algebra that is presented throughout the treatise.

Gabriel D. Villa Salvador

Departamento de Control Automático

CINVESTAV del I.P.N.

Ciudad de México

May, 2023

Introduction

Some people think that God is a geometer, a hard to prove statement. A more down to earth one, which goes along with Mother Nature, is: Mathematics is the greatest human mind creation that brings with, beautiful aesthetic feelings, as well as a better understanding of Nature.

Linear algebra, together with calculus, is the cornerstone in the mathematics curriculum of science and engineering. This might explain why there are many written books to approach those topics, which are included in a bachelor degree curriculum program. Taking this into account, it is natural to ask: Is it necessary to write another book on linear algebra? A general answer can go along the following lines. All branches of mathematics are in a continuous development, exploration, and review of new results and methods. This might justify why new linear algebra books or textbooks are needed all the time.

When writing this book, we kept in mind a couple of ideas. Firstly, to present our view about linear algebra and its learning; secondly, to provide a reference that might help students in their learning processes of this important branch of mathematics.

Usually, the discussion of a topic starts by clarifying which objects are going to be dealt with. Our case is not the exception, however, it would be very difficult to give a precise description of what linear algebra is all about.

To this respect, we want to point out that linear algebra could be thought as the part of mathematics that studies the equations

(1)AX=BandAX=λX,

where A, B, and X are matrices, and λ is a scalar.

Taking this into account, we could say that this book is developed from that point of view, that is, linear algebra is the study of equations in (1), adding three ingredients which we consider important in the mathematical activity: foundations, methods, and applications.

The starting point of the book was the writing of lecture notes that arose from teaching linear algebra courses during several years, particularly at the Autonomous University of Hidalgo State (UAEH), hence the discussion approach is influenced by the content of the curriculum in the bachelor’s degree program in Applied Mathematics, offered in UAEH. However, this book can be used in any engineering or science program, where one or two linear algebra courses are required.

Concerning the structure and approach of the topics contained in the book, we think it is appropriate to point out some ideas.

The first five chapters are thought as an introductory linear algebra course. In Chapter 1, we start by presenting hypothetical production models, which are discussed via a system of linear equations. The main idea behind this is to illustrate how linear equations could be used to solve real world problems. Continuing with this idea, in Chapter 2 we use again a hypothetical production model to motivate the definition of a matrix product. This approach to introduce the matrix product has its origin in the observation that for students it seems to be a little bit “artificial,” since the sum of matrices is defined by adding the corresponding entries. In Chapters 3, 4, and 5, basic geometric and algebraic properties of a finite-dimensional euclidean vector space are presented. The main topics in these chapters are the concepts and results related to linearly independent sets, dimension, and basis of a vector space, as well as general properties of a linear transformation.

The discussion level in Chapters 6 through 9 is a little bit more advanced. In Chapter 6, starting from a geometric approach, we introduce the determinant function as a means to represent area or volume of parallelograms or parallelepipeds, respectively, determined by vectors. Basic results of the theory of determinants are discussed. Also in this chapter we present a nice proof, credited to Whitford and Klamkin (1953), of the well-known Cramer’s rule.

Chapter 7 is the inspiration of the book. We want to mention a few ideas that originated the way the theory of eigenvalues and eigenvectors is discussed. In most linear algebra books, the approach to the theory of eigenvalues and eigenvectors follows the route of the characteristic polynomial and, when needed, the minimal polynomial is invoked. While using the minimal polynomial approach to...

Produkt-Details

• Erscheint lt. Verlag: 8.5.2023
• Reihe/Serie: De Gruyter Textbook
• Reihe/Serie: De Gruyter Textbook
• Zusatzinfo: 25 b/w ill.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Schlagworte: Eigen-Theorie • eigen theory • linear algebra • Linear algebra, eigen theory, linear equations, vector spaces, linear transformations. • Lineare Algebra • Lineare Gleichungen • Linear equations • linear transformations. • vector spaces • Vektorräume
• ISBN-10: 3-11-113614-0 / 3111136140
• ISBN-13: 978-3-11-113614-1 / 9783111136141