Linear Algebra E-Book

Tom M. Apostol

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 733

Veröffentlichungsjahr: 2014

Beschreibung

Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.

Leseprobe

CONTENTS

Preface

0. REVIEW OF PREREQUISITES

Part 1. Pre-calculus Prerequisites

0.1 Real numbers as points on a line

0.2 Pairs of real numbers as points in a plane

0.3 Polar coordinates

0.4 Complex numbers

0.5 Definition and algebraic properties of complex numbers

0.6 Complex numbers as an extension of real numbers

0.7 The imaginary unit i

0.8 Exercises

0.9 Geometric interpretation. Modulus and argument

0.10 Complex conjugates

0.11 Exercises

0.12 Mathematical induction

0.13 Exercises

0.14 Necessary and sufficient conditions

Part 2. Calculus Prerequisites

0.15 The concept of derivative

0.16 Basic properties of derivatives

0.17 Derivatives of some elementary functions

0.18 Velocity and acceleration

0.19 The area problem and the history of integral calculus

0.20 Integration as a process for producing new functions

0.21 Basic properties of the integral

0.22 The exponential function

0.23 Complex exponentials

0.24 Polar form of complex numbers

0.25 Power series and series of functions

0.26 Exercises

1. VECTOR ALGEBRA

1.1 Historical introduction

1.2 The vector space of n-tuples of real numbers

1.3 Geometric interpretation for n ≤ 3

1.4 Exercises

1.5 The dot product

1.6 Length or norm of a vector

1.7 Orthogonality of vectors

1.8 Exercises

1.9 Projections. Angle between vectors in n-space

1.10 The unit coordinate vectors

1.11 Exercises

1.12 The linear span of a finite set of vectors

1.13 Linear independence

1.14 Bases

1.15 Exercises

1.16 The vector space Cn of n-tuples of complex numbers

1.17 Exercises

2. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY

2.1 Introduction

2.2 Lines in n-space

2.3 Some simple properties of straight lines in Rn

2.4 Lines and vector-valued functions in n-space

2.5 Lines in 3-space and in 2-space

2.6 Exercises

2.7 Planes in Euclidean n-space

2.8 Planes and vector-valued functions

2.9 Exercises

2.10 The cross product of two vectors in R3

2.11 The cross product expressed as a determinant

2.12 Exercises

2.13 The scalar triple product

2.14 Cramer’s rule for solving a system of three linear equations

2.15 Exercises

2.16 Normal vectors to planes in R3

2.17 Linear Cartesian equations for planes in R3

2.18 Exercises

2.19 The conic sections

2.20 Eccentricity of conic sections

2.21 Polar equations for conic sections

2.22 Exercises

2.23 Cartesian equation for a general conic

2.24 Conic sections symmetric about the origin

2.25 Cartesian equations for the ellipse and the hyperbola in standard position

2.26 Cartesian equations for the parabola

2.27 Exercises

2.28 Miscellaneous exercises on conic sections

3. LINEAR SPACES

3.1 Introduction

3.2 Axiomatic definition of a linear space

3.3 Examples of linear spaces

3.4 Elementary consequences of the axioms

3.5 Exercises

3.6 Subspaces of a linear space

3.7 Dependent and independent sets in a linear space

3.8 Bases and dimension

3.9 Components

3.10 Exercises

3.11 Inner products, Euclidean spaces. Norms

3.12 Orthogonality in a Euclidean space

3.13 Exercises

3.14 Construction of orthogonal sets. The Gram-Schmidt process

3.15 Orthogonal complements. Projections

3.16 Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace

3.17 Exercises

4. LINEAR TRANSFORMATIONS AND MATRICES

4.1 Linear transformations

4.2 Null space and range

4.3 Nullity and rank

4.4 Exercises

4.5 Algebraic operations on linear transformations

4.6 Inverses

4.7 One-to-one linear transformations

4.8 Exercises

4.9 Linear transformations with prescribed values at the elements of a basis

4.10 Matrix representations of linear transformations

4.11 Construction of a matrix representation in diagonal form

4.12 Exercises

4.13 Linear spaces of matrices

4.14 Isomorphism between linear transformations and matrices

4.15 Multiplication of matrices

4.16 Exercises

4.17 Applications to systems of linear equations

4.18 Computation techniques. The Gauss-Jordan method

4.19 Inverses of square matrices

4.20 Exercises

4.21 Miscellaneous exercises on matrices

5. DETERMINANTS

5.1 Introduction

5.2 Motivation for the choice of axioms for a determinant function

5.3 A set of axioms for a determinant function

5.4 The determinant of a diagonal matrix

5.5 The determinant of an upper triangular matrix

5.6 Computation of any determinant by the Gauss-Jordan process

5.7 Uniqueness of the determinant function

5.8 Exercises

5.9 Multilinearity of determinants

5.10 Applications of multilinearity

5.11 The product formula for determinants

5.12 The determinant of the inverse of a nonsingular matrix

5.13 Determinants and independence of vectors

5.14 The determinant of a block-diagonal matrix

5.15 Exercises

5.16 Expansion formulas by cofactors

5.17 The cofactor matrix

5.18 Cramer’s rule

5.19 Expansion formulas by minors

5.20 Exercises

5.21 Existence of the determinant function

5.22 Miscellaneous exercises on determinants

6. EIGENVALUES AND EIGENVECTORS

6.1 Linear transformations with diagonal matrix representations

6.2 Eigenvalues and eigenvectors of a linear transformation

6.3 Linear independence of eigenvectors corresponding to distinct eigenvalues

6.4 Exercises

6.5 The finite-dimensional case

6.6 The triangularization theorem

6.7 Characteristic polynomials

6.8 Calculation of eigenvalues and eigenvectors in the finite-dimensional case

6.9 The product and sum of the roots of a characteristic polynomial

6.10 Exercises

6.11 Matrices representing the same linear transformation. Similar matrices

6.12 Exercises

6.13 The Cayley-Hamilton theorem

6.14 Exercises

6.15 The Jordan normal form

6.16 Miscellaneous exercises on eigenvalues and eigenvectors

7. EIGENVALUES OF OPERATORS ACTING ON EUCLIDEAN SPACES

7.1 Eigenvalues and inner products

7.2 Hermitian and skew-Hermitian transformations

7.3 Orthogonality of eigenvectors corresponding to distinct eigenvalues

7.4 Exercises

7.5 Existence of an orthonormal set of eigenvectors for Hermitian and skew-Hermitian operators acting on finite-dimensional spaces

7.6 Matrix representations for Hermitian and skew-Hermitian operators

7.7 Hermitian and skew-Hermitian matrices. The adjoint of a matrix

7.8 Diagonalization of a Hermitian or skew-Hermitian matrix

7.9 Unitary matrices. Orthogonal matrices

7.10 Exercises

7.11 Quadratic forms

7.12 Reduction of a real quadratic form to a diagonal form

7.13 Applications to conic sections

7.14 Exercises

7.15 Positive definite quadratic forms

*7.16 Eigenvalues of a symmetric transformation obtained as values of its quadratic form

*7.17 Extremal properties of eigenvalues of a symmetric transformation

*7.18 The finite-dimensional case

7.19 Unitary transformations

7.20 Exercises

*7.21 Examples of symmetric and skew-symmetric operators acting on function spaces

7.22 Exercises

8. APPLICATIONS TO LINEAR DIFFERENTIAL EQUATIONS

8.1 Introduction

8.2 Review of results concerning linear differential equations of first and second orders

8.3 Exercises

8.4 Linear differential equations of order n

8.5 The existence-uniqueness theorem

8.6 The dimension of the solution space of a homogeneous linear differential equation

8.7 The algebra of constant-coefficient operators

8.8 Determination of a basis of solutions for linear equations with constant coefficients by factorization of operators

8.9 Exercises

8.10 The relation between the homogeneous and nonhomogeneous equations

8.11 Determination of a particular solution of the nonhomogeneous equation. The method of variation of parameters

8.12 Nonsingularity of the Wronskian matrix of n independent solutions of a homogeneous linear equation

8.13 Special methods for determining a particular solution of the nonhomogeneous equation. Reduction to a system of first-order linear equations

8.14 The annihilator method for determining a particular solution of the nonhomogeneous equation

8.15 Exercises

9. APPLICATIONS TO SYSTEMS OF DIFFERENTIAL EQUATIONS

9.1 Introduction

9.2 Calculus of matrix functions

9.3 Infinite series of matrices. Norms of matrices

9.4 Exercises

9.5 The exponential matrix

9.6 The differential equation satisfied by etA

9.8 The law of exponents for exponential matrices

9.9 Existence and uniqueness theorems for homogeneous linear systems with constant coefficients

9.10 Calculating etA in special cases

9.11 Exercises

9.12 Putzer’s method for calculating etA

9.13 Alternate methods for calculating etA in special cases

9.14 Exercises

9.15 Nonhomogeneous linear systems with constant coefficients

9.16 Exercises

9.18 A power series method for solving homogeneous linear systems

9.19 Exercises

10. THE METHOD OF SUCCESSIVE APPROXIMATIONS

10.1 Introduction

10.3 Convergence of the sequence of successive approximations

10.4 The method of successive approximations applied to first-order nonlinear systems

10.5 Proof of an existence-uniqueness theorem for first-order nonlinear systems

10.6 Exercises

*10.7 Successive approximations and fixed points of operators

*10.8 Normed linear spaces

*10.9 Contraction operators

*10.10 Fixed-point theorem for contraction operators

*10.11 Applications of the fixed-point theorem

Answers to Exercises

Index

A NOTE TO THE READER

This book has been electronically reproduced from digital information stored at John Wiley & Sons, Inc. We are pleased that the use of this new technology will enable us to keep works of enduring scholarly value in print as long as there is a reasonable demand for them. The content of this book is identical to previous printings.

This text is printed on acid-free paper.

Copyright © 1997 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

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Library of Congress Cataloging in Publication Data:

Apostol, Tom M.

Linear algebra : a first course, with applications to differential equations / Tom M. Apostol.

p. cm.

“A Wiley-Interscience publication.”

ISBN 0-471-17421-1 (cloth : acid-free paper)

1. Algebra, Linear. 2. Differential equations. I. Title.

QA184.A66 1997

512′.5—dc21      96-37131

CIP

PREFACE

For many years the author has been urged to develop a text on linear algebra based on material in the second edition of his two-volume Calculus, which presents calculus of functions of one or more variables, integrated with differential equations, infinite series, linear algebra, probability, and numerical analysis. To some extent this was done by others when the two Calculus volumes were translated into Italian and divided into three volumes,* the second of which contained the material on linear algebra. The present text is designed to be independent of the Calculus volumes.

To accommodate a variety of backgrounds and interests, this text begins with a review of prerequisites (Chapter 0). The review is divided into two parts: pre-calculus prerequisites, needed to understand the material in Chapters 1 through 7, and calculus prerequisites, needed for Chapters 8 through 10. Chapters 1 and 2 introduce vector algebra in n-space with applications to analytic geometry. These two chapters provide motivation and concrete examples to illustrate the more abstract treatment of linear algebra presented in Chapters 3 through 7.

Chapter 3 discusses linear spaces, subspaces, linear independence, bases and dimension, inner products, orthogonality, and the Gram-Schmidt process. Chapter 4 introduces linear transformations and matrices, with applications to systems of linear equations. Chapter 5 is devoted to determinants, which are introduced axiomatically through their properties. The treatment is somewhat simpler than that given in the author’s Calculus. Chapter 6 treats eigenvalues and eigenvectors, and includes the triangularization theorem, which is used to deduce the Cayley-Hamilton theorem. There is also a brief section on the Jordan normal form. Chapter 7 continues the discussion of eigenvalues and eigenvectors in the setting of Euclidean spaces, with applications to quadratic forms and conic sections.

In Chapters 3 through 7, calculus concepts occur only occasionally in some illustrative examples, or in some of the exercises; these are clearly identified and can be omitted or postponed without disrupting the continuity of the text. This part of the text is suitable for a first course in linear algebra not requiring a calculus prerequisite. However, the level of presentation is more appropriate for readers who have acquired some degree of mathematical sophistication in a course such as elementary calculus or finite mathematics.

Chapters 8, 9, and 10 definitely require a calculus background. Chapter 8 applies linear algebra concepts to linear differential equations of order n, with special emphasis on equations with constant coefficients. Chapter 9 uses matrix calculus to discuss systems of differential equations. This chapter focuses on the exponential matrix, whose properties are derived by an interplay between linear algebra and matrix calculus. Chapter 10 treats existence and uniqueness theorems for systems of differential equations, using Picard’s method of successive approximations, which is also cast in the language of contraction operators.

Although most of the material in this book was extracted from the author’s Calculus, some topics have been revised or rearranged, and some new material and new exercises have been added.

This textbook can be used by first- or second-year students in college, and it can also be of interest to more mature individuals, who may have studied mathematics many years ago without learning linear algebra, and who now wish to learn the basic concepts without undue emphasis on abstraction or formalization.

TOM M. APOSTOL

California Institute of Technology

*Calcolo, Volume primo: Analisi 1; Volume Secondo: Geometria; Volume Terzo: Analisi 2. Published by Editore Boringhieri, 1977.

0

REVIEW OF PREREQUISITES

Part 1 of this chapter summarizes some pre-calculus prerequisites for this book—facts about real numbers, rectangular coordinates, complex numbers, and mathematical induction. Part 2 does the same for calculus prerequisites. Chapters 1 and 2, which deal with vector algebra and its applications to analytic geometry, do not require calculus as a prerequisite. These two chapters provide motivation and concrete examples to illustrate the abstract treatment of linear algebra that begins with Chapter 3. In Chapters 3 through 7, calculus concepts occur only occasionally in some illustrative examples, or in some exercises; these are clearly identified and can be omitted or postponed without disrupting the continuity of the text.

Although calculus and linear algebra are independent subjects, some of the most striking applications of linear algebra involve calculus concepts—integrals, derivatives, and infinite series. Familiarity with one-variable calculus is essential to understand these applications, especially those referring to differential equations presented in the last three chapters. At the same time, the use of linear algebra places some aspects of differential equations in a natural setting and helps increase understanding.

Part 1. Pre-calculus Prerequisites

0.1 Real numbers as points on a line

Real numbers can be represented geometrically as points on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 0.1. This choice determines the scale, or unit of measure. If one adopts an appropriate set of axioms for Euclidean geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number. For this reason, the line is usually called the real line or the real axis. We often speak of the point x rather than the point corresponding to the real number x. The set of all real numbers is denoted by R.

FIGURE 0.1 Real numbers represented geometrically on a line.

If x < y, point x lies to the left of y as shown in Figure 0.1. Each positive real number x lies at a distance x to the right of zero. A negative real number x is represented by a point located at a distance |x| to the left of zero.

0.2 Pairs of real numbers as points in a plane

Points in a plane can be represented by pairs of real numbers. Two perpendicular reference lines in the plane are chosen, a horizontal x axis and a vertical y axis. Their point of intersection, denoted by 0, is called the origin. On the x axis a convenient point is chosen to the right of 0 to represent 1; its distance from 0 is called the unit distance. Vertical distances along the y axis are usually measured with the same unit distance. Each point in the plane is assigned a pair of numbers, called its coordinates, which tell us how to locate the point. Figure 0.2 illustrates some examples. The point with coordinates (3, 2) lies three units to the right of the y axis and two units above the x axis. The number 3 is called the x coordinate or abscissa of the point, and 2 is its y coordinate or ordinate. Points to the left of the y axis have a negative abscissa; those below the x axis have a negative ordinate. The coordinates of a point, as just defined, are called its Cartesian coordinates in honor of René Descartes (1596–1650), one of the founders of analytic geometry.

The procedure for locating points in space is analogous. We take three mutually perpendicular lines in space intersecting at a point (the origin). These lines determine three mutually perpendicular planes, and each point in space can be completely described by specifying, with appropriate regard for signs, the distances from these planes. We shall discuss three-dimensional Cartesian coordinates in a later chapter; for the present we confine our attention to the two-dimensional case.

FIGURE 0.2 Points in the plane represented by pairs of real numbers.

A geometric figure, such as a curve in the plane, is a collection of points satisfying one or more special conditions. By expressing these conditions in terms of the coordinates x and y we obtain one or more relations (equations or inequalitites) that characterize the figure in question. For example, consider a circle of radius r with its center at the origin, as shown in Figure 0.3.

Let (x, y) denote the coordinates of an arbitrary point P on this circle. The line segment OP is the hypotenuse of a right triangle whose legs have lengths || and | and, hence, by the theorem of Pythagoras, we have