TAILIEUCHUNG - Exercises and Problems in Linear Algebra

This collection of exercises is designed to provide a framework for discussion in a junior level linear algebra class such as the one I have conducted fairly regularly at Portland State University. There is no assigned text. Students are free to choose their own sources of information. Students are encouraged to find books, papers, and web sites whose writing style they find congenial, whose emphasis matches their interests, and whose price fits their budgets. The short introductory background section in these exercises, which precede each assignment, are intended only to fix notation and provide “official” definitions and statements of important theorems for the exercises and problems which follow. | Exercises and Problems in Linear Algebra John M. Erdman Portland State University Version July 13 2014 2010 c John M. Erdman E-mail address erdman@ Contents PREFACE vii Part 1. MATRICES AND LINEAR EQUATIONS 1 Chapter 1. SYSTEMS OF LINEAR EQUATIONS 3 . Background 3 . Exercises 4 . Problems 7 . Answers to Odd-Numbered Exercises 8 Chapter 2. ARITHMETIC OF MATRICES 9 . Background 9 . Exercises 10 . Problems 12 . Answers to Odd-Numbered Exercises 14 Chapter 3. ELEMENTARY MATRICES DETERMINANTS 15 . Background 15 . Exercises 17 . Problems 22 . Answers to Odd-Numbered Exercises 23 Chapter 4. VECTOR GEOMETRY IN Rn 25 . Background 25 . Exercises 26 . Problems 28 . Answers to Odd-Numbered Exercises 29 Part 2. VECTOR SPACES 31 Chapter 5. VECTOR SPACES 33 . Background 33 . Exercises 34 . Problems 37 . Answers to Odd-Numbered Exercises 38 Chapter 6. SUBSPACES 39 . Background 39 . Exercises 40 . Problems 44 . Answers to Odd-Numbered Exercises 45 Chapter 7. LINEAR INDEPENDENCE 47 . Background 47 . Exercises 49 iii iv CONTENTS . Problems 51 . Answers to Odd-Numbered Exercises 53 Chapter 8. BASIS FOR A VECTOR SPACE 55 . Background 55 . Exercises 56 . Problems 57 . Answers to Odd-Numbered Exercises 58 Part 3. LINEAR MAPS BETWEEN VECTOR SPACES 59 Chapter 9. LINEARITY 61 . Background 61 . Exercises 63 . Problems 67 . Answers to Odd-Numbered Exercises 70 Chapter 10. LINEAR MAPS BETWEEN EUCLIDEAN SPACES 71 . Background 71 . Exercises 72 . Problems 74 . Answers to Odd-Numbered Exercises 75 Chapter 11. PROJECTION OPERATORS 77 . Background 77 . Exercises 78 . Problems 79 . Answers to Odd-Numbered Exercises 80 Part 4. SPECTRAL THEORY OF VECTOR SPACES 81 Chapter 12. EIGENVALUES AND EIGENVECTORS 83 . Background 83 . Exercises 84 . Problems 85 . Answers to Odd-Numbered Exercises 86 Chapter 13. DIAGONALIZATION OF MATRICES 87 . .

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