SOLUTIONS MANUAL for Linear Algebra A Pathway to Abstract Mathematics 3rd Edition by G. Viglino

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LINEAR ALGEBRA A Pathway to Abstract Mathematics 3rd Edition G. Viglino

Ramapo College of New Jersey January 2018

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Contents

CONTENTS CHAPTER 1

MATRICES AND SYSTEMS OF LINEAR EQUATION

1.1

Systems of Linear Equations

1

1.2

Consistent and Inconsistent Systems of Equations

14

Chapter Summary

CHAPTER 2

27

VECTOR SPACES

2.1

Vectors in the plane, and Beyond

31

2.2

Abstract Vector Spaces

40

2.3

Properties of Vector Spaces

51

2.4

Subspaces

59

2.5

Lines and Planes

68

Chapter Summary

CHAPTER 3

75

BASES AND DIMENSION

3.1

Spanning Sets

77

3.2

Linear Independence

86

3.3

Bases

94

Chapter Summary

109

CHAPTER4

LINEARITY

4.1

Linear Transformations

111

4.2

Image and Kernel

124

4.3

Isomorphisms

135

Chapter Summary

CHAPTER 5

149

MATRICES AND LINEAR MAPS

5.1

Matrix Multiplication

151

5.2

Invertible Matrices

165

5.3

Matrix Representation of Linear Maps

178

5.4

Change of Basis

191

Chapter Summary

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202

Contents

CHAPTER 6

DETERMINANTS AND EIGENVECTORS

6.1

Determinants

205

6.2

Eigenspaces

218

6.3

Diagonalization

233

6.4

Applications

246

6.5

Markov Chains

259

Chapter Summary

CHAPTER 7

275

INNER PRODUCT SPACES

7.1

Dot Product

279

7.2

Inner Product

292

7.3

Orthogonality

301

7.4.

The Spectral Theorem

315

Chapter Summary

Appendix A

Principle of Mathematical Induction

Appendix B

Solutions to Check Your Understanding Boxes

Appendix C

Answers to Selected Odd-Numbered Exercises

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307

Preface

PREFACE There is no mathematical ramp that will enable you to continuously inch your way higher and higher in mathematics. The climb calls for a ladder consisting of discrete steps designed to take you from one mathematical level to another. You are about to take an important step on that ladder, one that will take you to a plateau where mathematical abstraction abounds. Linear algebra rests on a small number of axioms (accepted rules, or “laws”), upon which a beautiful and practical theory emerges.

Technology can be used to reduce the time needed to perform essential but routine tasks. We feature the TI-84+ calculator, but any graphing utility or Computer Algebraic System will do. The real value of whatever technological tool you use is that it will free you to spend more time on the development and comprehension of the theory and its applications. In any event, if you haven’t already discovered in other courses:

MATHEMATICS DOES NOT RUN ON BATTERIES Systems of linear equations are introduced and analyzed in Chapter 1. Graphing utilities can be used to solve such systems, but understanding what those solutions represent plays a dominant role throughout the text.

We begin Chapter 2 by sowing the seeds for vector spaces in the fertile real number field, where they soon blossom into the concept of an abstract vector. The remainder of Chapter 2 and all of Chapter 3 are dedicated to a study of vector spaces in isolation. Functions from one vector space to another which, in a sense, respect the algebraic structure of those spaces are investigated in Chapters 4 and 5. The sixth chapter focuses on Eigenvalues and Eigenvectors, along with some of their important applications.

The first six chapters may provide a full plate for most one-semester courses. If not, then Chapter 7 (on inner product spaces) is offered for dessert.

We have made every effort to provide a leg-up for the step you are about to take. Our primary goal was to write a readable book, without compromising mathematical integrity. Along the way, you will encounter numerous Check Your Understanding boxes designed to challenge your understanding of each newly introduced concept. Complete solutions to the problems in those boxes appear in Appendix B, but please don’t be in too much of a hurry to look at our solutions. You should TRY to solve the problems on your own, for it is only through ATTEMPTING to solve a problem that one grows mathematically. In the words of Descartes:

We never understand a thing so well, and make it our own, when we learn it from another, as when we have discovered it for ourselves.

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