Linear Algebra A First Course with Applications to Differential Equations

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ISBN: 9780471174219 | 0471174211
Cover: Hardcover
Copyright: 7/15/1997

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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.

Preface

0. REVIEW OF PREREQUISITES

(24)

Part 1. Pre-calculus Prerequisites

(13)

0.1 Real numbers as points on a line

(1)

0.2 Pairs of real numbers as points in a plane

(1)

0.3 Polar coordinates

(1)

0.4 Complex numbers

(1)

0.5 Definition and algebraic properties of complex numbers

(1)

0.6 Complex numbers as an extension of real numbers

(1)

0.7 The imaginary unit i

(2)

0.8 Exercises

(1)

0.9 Geometric interpretation. Modulus and argument

(2)

0.10 Complex conjugates

(1)

0.11 Exercises

(1)

0.12 Mathematical induction

(2)

0.13 Exercises

(1)

0.14 Necessary and sufficient

(1)

Part 2. Calculus Prerequisites

(11)

0.15 The concept of derivative

(1)

0.16 Basic properties of derivatives

(1)

0.17 Derivatives of some elementary functions

(1)

0.18 Velocity and acceleration

(1)

0.19 The area problem and the history of integral calculus

(1)

0.20 Integration as a process for producing new functions

(1)

0.21 Basic properties of the integral

(1)

0.22 The exponential function

(1)

0.23 Complex exponentials

(1)

0.24 Polar form of complex numbers

(1)

0.25 Power series and series of functions

(2)

0.26 Exercises

(1)

1. VECTOR ALGEBRA

(26)

1.1 Historical introduction

(1)

1.2 The vector space of n-tuples of real numbers

(2)

1.3 Geometric interpretation for n less than 3

(2)

1.4 Exercises

(1)

1.5 The dot product

(2)

1.6 Length or norm of a vector

(2)

1.7 Orthogonality of vectors

(1)

1.8 Exercises

(1)

1.9 Projections. Angle between vectors in n-space

(2)

1.10 The unit coordinate vectors

(2)

1.11 Exercises

(1)

1.12 The linear span of a finite set of vectors

(1)

1.13 Linear independence

(3)

1.14 Bases

(1)

1.15 Exercises

(1)

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

(2)

1.17 Exercises

(1)

2. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY

(40)

2.1 Introduction

(11)

2.2 Lines in n-space

(1)

12.3 Some simple properties of straight lines in Rn

(1)

2.4 Lines and vector-valued functions in n-space

(1)

2.5 Lines in 3-space in 2-space

(2)

2.6 Exercises

(1)

2.7 Planes in Euclidean n-space

(3)

2.8 Planes and vector-valued functions

(1)

2.9 Exercises

(1)

2.10 The cross product of two vectors in R3

(3)

2.11 The cross product expressed as a determinant

(1)

2.12 Exercises

(1)

2.13 The scalar triple product

(2)

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

(1)

2.15 Exercises

(2)

2.16 Normal vectors to planes in R3

(1)

2.17 Linear Cartesian equations for planes in R3

(2)

2.18 Exercises

(1)

2.19 The conic sections

(2)

2.20 Eccentricity of conic sections

(2)

2.21 Polar equations for conic sections

(1)

2.22 Exercises

(1)

2.23 Cartesian equation for a general conic

(1)

2.24 Conic sections symmetric about the origin

(1)

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

(2)

2.26 Cartesian equations for the parabola

(1)

2.27 Exercises

(2)

2.28 Miscellaneous exercises on conic sections

(1)

3. LINEAR SPACES

(28)

3.1 Introduction

(1)

3.2 Axiomatic definition of a linear space

(1)

3.3 Examples of linear spaces

(2)

3.4 Elementary consequences of the axioms

(1)

3.5 Exercises

(1)

3.6 Subspaces of a linear space

(2)

3.7 Dependent and independent sets in a linear space

(2)

3.8 Bases and dimension

100

(2)

3.9 Components

102

(1)

3.10 Exercises

102

(1)

3.11 Inner products, Euclidean spaces. Norms

103

(4)

3.12 Orthogonality in a Euclidean space

107

(2)

3.13 Exercises

109

(2)

3.14 Construction of orthogonal sets. The Gram-Schmidt process

111

(4)

3.15 Orthogonal complements. Projections

115

(1)

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

116

(2)

3.17 Exercises

118

(1)

4. LINEAR TRANSFORMATIONS AND MATRICES

119

(42)

4.1 Linear transformations

119

(2)

4.2 Null space and range

121

(1)

4.3 Nullity and rank

122

(1)

4.4 Exercises

123

(1)

4.5 Algebraic operations on linear transformations

124

(2)

4.6 Inverses

126

(2)

4.7 One-to-one linear transformations

128

(2)

4.8 Exercises

130

(1)

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

131

(1)

4.10 Matrix representations of linear transformations

132

(5)

4.11 Construction of a matrix representation in diagonal form

137

(2)

4.12 Exercises

139

(1)

4.13 Linear spaces of matrices

140

(2)

4.14 Isomorphism between linear transformations and matrices

142

(1)

4.15 Multiplication of matrices

143

(3)

4.16 Exercises

146

(1)

4.17 Applications to systems of linear equations

147

(3)

4.18 Computation techniques. The Gauss-Jordan method

150

(4)

4.19 Inverses of square matrices

154

(4)

4.20 Exercises

158

(1)

4.21 Miscellaneous exercises on matrices

159

(2)

5. DETERMINANTS

161

(26)

5.1 Introduction

161

(1)

5.2 Motivation for the choice of axioms for a determinant function

162

(1)

5.3 A set of axioms for a determinant function

163

(1)

5.4 The determinant of a diagonal matrix

164

(1)

5.5 The determinant of an upper triangular matrix

165

(1)

5.6 Computation of any determinant by the Gauss-Jordan process

166

(1)

5.7 Uniqueness of the determinant function

166

(1)

5.8 Exercises

167

(1)

5.9 Multilinearity of determinants

168

(2)

5.10 Applications of multilinearity

170

(1)

5.11 The product formula for determinants

171

(1)

5.12 The determinant of the inverse of a nonsingular matrix

172

(1)

5.13 Determinants and independence of vectors

173

(1)

5.14 The determinant of a block-diagonal matrix

173

(1)

5.15 Exercises

174

(1)

5.16 Expansion formulas by cofactors

175

(2)

5.17 The cofactor matrix

177

(1)

5.18 Cramer's rule

178

(1)

5.19 Expansion formulas by minors

179

(2)

5.20 Exercises

181

(1)

5.21 Existence of the determinant function

182

(2)

5.22 Miscellaneous exercises on determinants

184

(3)

6. EIGENVALUES AND EIGENVECTORS

187

(30)

6.1 Linear transformations with diagonal matrix representations

187

(1)

6.2 Eigenvalues and eigenvectors of a linear transformation

188

(2)

6.3 Linear independence of eigenvectors corresponding to distinct eigenvalues

190

(1)

6.4 Exercises

191

(1)

6.5 The finite-dimensional case

192

(1)

6.6 The triangularization theorem

193

(3)

6.7 Characteristic polynomials

196

(2)

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

198

(3)

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

201

(1)

6.10 Exercises

202

(1)

6.11 Matrices representing the same linear transformation. Similar matrices

203

(4)

6.12 Exercises

207

(1)

6.13 The Cayley-Hamilton theorem

208

(3)

6.14 Exercises

211

(1)

6.15 The Jordan normal form

211

(3)

6.16 Miscellaneous exercises on eigenvalues and eigenvectors

214

(3)

7. EIGENVALUES OF OPERATORS ACTING ON EUCLIDEAN SPACES

217

(30)

7.1 Eigenvalues and inner products

217

(1)

7.2 Hermitian and skew-Hermitian transformations

218

(1)

7.3 Orthogonality of eigenvectors corresponding to distinct eigenvalues

219

(1)

Exercises

219

(1)

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

220

(1)

7.6 Matrix representations for Hermitian and skew-Hermitian operators

221

(1)

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

222

(1)

7.8 Diagonalization of a Hermitian or skew-Hermitian matrix

223

(1)

7.9 Unitary matrices. Orthogonal matrices

224

(1)

7.10 Exercises

225

(2)

7.11 Quadratic forms

227

(2)

7.12 Reduction of a real quadratic form to a diagonal form

229

(2)

7.13 Applications to conic sections

231

(4)

7.14 Exercises

235

(1)

7.15 Positive definite quadratic forms

235

(1)

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

236

(2)

*7.17 Extremal properties of eigenvalues of a symmetric transformation

238

(1)

*7.18 The finite-dimensional case

239

(1)

7.19 Unitary transformations

240

(2)

7.20 Exercises

242

(1)

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

243

(2)

7.22 Exercises

245

(2)

8. APPLICATIONS TO LINEAR DIFFERENTIAL EQUATIONS

247

(26)

8.1 Introduction

247

(1)

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

248

(2)

8.3 Exercises

250

(1)

8.4 Linear differential of order n

251

(1)

8.5 The existence-uniqueness theorem

252

(1)

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

252

(1)

8.7 The algebra of constant-coefficient operators

253

(2)

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

255

(4)

Exercises

259

(1)

8.10 The relation between the homogeneous and nonhomogeneous equations

260

(1)

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

261

(4)

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

265

(1)

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

266

(1)

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

267

(3)

8.15 Exercises

270

(3)

9. APPLICATIONS TO SYSTEMS OF DIFFERENTIAL EQUATIONS

273

(30)

9.1 Introduction

273

(2)

9.2 Calculus of matrix functions

275

(1)

9.3 Infinite series of matrices. Norms of matrices

276

(1)

9.4 Exercises

277

(1)

9.5 The exponential matrix

278

(1)

9.6 The differential equation satisfied by etA

279

(1)

9.7 Uniqueness theorem for the matrix differential equation F1(t) = AF(t)

279

(2)

9.8 The law of exponents for exponential matrices

281

(1)

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

281

(1)

9.10 Calculating etA in special cases

282

(5)

9.11 Exercises

287

(1)

9.12 Putzer's method for calculating etA

288

(3)

9.13 Alternate methods for calculating etA in special cases

291

(2)

9.14 Exercises

293

(1)

9.15 Nonhomogeneous linear systems with constant coefficients

294

(2)

9.16 Exercises

296

(2)

9.17 The general linear system Y1(t) = P(t)Y(t) + Q(t)

298

(3)

9.18 A power series method for solving homogeneous linear systems

301

(1)

9.19 Exercises

302

(1)

10. THE METHOD OF SUCCESSIVE APPROXIMATIONS

303

(18)

10.1 Introduction

303

(1)

10.2 Application to the homogeneous linear system Y1(t) = A(t)Y(t)

303

(1)

10.3 Convergence of the sequence of successive approximations

304

(4)

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

308

(1)

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

309

(2)

10.6 Exercises

311

(1)

*10.7 Successive approximations and fixed points of operators

312

(1)

*10.8 Normed linear spaces

313

(1)

*10.9 Contraction operators

314

(1)

*10.10 Fixed-point theorem for contraction operators

315

(2)

*10.11 Applications of the fixed-point theorem

317

(4)

Answers to Exercises

321

(22)

Index

343

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Linear Algebra A First Course with Applications to Differential Equations

Summary

Author Biography

Table of Contents

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