Linear Algebra

James R. Kirkwood, Bessie H. Kirkwood

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9781498776851
• eBook ISBN: 9781498776875
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Linear Algebra, James R. Kirkwood and Bessie H. Kirkwood, 978-1-4987-7685-1, K29751

Shelving Guide: Mathematics

This text has a major focus on demonstrating facts and techniques of linear systems that will be invaluable in higher mathematics and related fields. A linear algebra course has two major audiences that it must satisfy. It provides an important theoretical and computational tool for nearly every discipline that uses mathematics. It also provides an introduction to abstract mathematics.

This book has two parts. Chapters 1–7 are written as an introduction. Two primary goals of these chapters are to enable students to become adept at computations and to develop an understanding of the theory of basic topics including linear transformations. Important applications are presented.

Part two, which consists of Chapters 8–14, is at a higher level. It includes topics not usually taught in a first course, such as a detailed justification of the Jordan canonical form, properties of the determinant derived from axioms, the Perron–Frobenius theorem and bilinear and quadratic forms.

Though users will want to make use of technology for many of the computations, topics are explained in the text in a way that will enable students to do these computations by hand if that is desired.

Key features include:

• Chapters 1–7 may be used for a first course relying on applications
• Chapters 8–14 offer a more advanced, theoretical course
• Definitions are highlighted throughout
• MATLAB® and R Project tutorials in the appendices
• Exercises span a range from simple computations to fairly direct abstract exercises
• Historical notes motivate the presentation

Chapter 1

Matrices

Linear algebra is the branch of mathematics that deals with vector spaces and linear transformations. For someone just beginning their study of linear algebra, that is probably a meaningless statement. It does, however, convey the idea that there are two concepts in our study that will be of utmost importance; namely, vector spaces and linear transformations. A primary tool in our study of these topics will be matrices. In this chapter we give the rules that govern matrix algebra.

Section 1.1 Matrix Arithmetic

The term “matrix” was first used in a mathematical sense by James Sylvester in 1850 to denote a rectangular array of numbers from which determinants could be formed. The idea of using arrays to solve systems of linear equations goes as far back as the Chinese circa 300 BC.

A matrix is...

Table of contents

• Cover
• Half-Title
• Series
• Title
• Copyright
• Contents
• Preface
• 1 Matrices
• 2 Systems of Linear Equations
• 3 Vector Spaces
• 4 Linear Transformations
• 5 Eigenvalues and Eigenvectors
• 6 Inner Product Spaces
• 7 Linear Functionals, Dual Spaces, and Adjoint Operators
• 8 Two Decompositions of a Matrix
• 9 Determinants
• 10 Jordan Canonical Form
• 11 Applications of the Jordan Canonical Form
• 12 The Perron–Frobenius Theorem
• 13 Bilinear Forms
• 14 Introduction to Tensor Product
• Appendix I: A Brief Guide to MATLAB
• Appendix II: R for Linear Algebra
• Answers to Selected Exercises
• Index