Introduction to Linear Algebra

Description

Linear algebra provides the essential mathematical tools to tackle all the problems in Science. Introduction to Linear Algebra is primarily aimed at students in applied fields (e.g. Computer Science and Engineering), providing them with a concrete, rigorous approach to face and solve various types of problems for the applications of their interest. This book offers a straightforward introduction to linear algebra that requires a minimal mathematical background to read and engage with. Features Presented in a brief, informative and engaging style Suitable for a wide broad range of undergraduates Contains many worked examples and exercises Cover Half Title Title Page Copyright Page Contents Preface Chapter 1: Introduction to Linear Systems 1.1. LINEAR SYSTEMS: FIRST EXAMPLES 1.2. MATRICES 1.3. MATRICES AND LINEAR SYSTEMS 1.4. THE GAUSSIAN ALGORITHM 1.5. EXERCISES WITH SOLUTIONS 1.6. SUGGESTED EXERCISES Chapter 2: Vector Spaces 2.1. INTRODUCTION: THE SET OF REAL NUMBERS 2.2. THE VECTOR SPACE RN AND THE VECTOR SPACE OF MATRICES 2.3. VECTOR SPACES 2.4. SUBSPACES 2.5. EXERCISES WITH SOLUTIONS 2.6. SUGGESTED EXERCISES Chapter 3: Linear Combination and Linear Independence 3.1. LINEAR COMBINATIONS AND GENERATORS 3.2. LINEAR INDEPENDENCE 3.3. EXERCISES WITH SOLUTIONS 3.4. SUGGESTED EXERCISES Chapter 4: Basis and Dimension 4.1. BASIS: DEFINITION AND EXAMPLES 4.2. THE CONCEPT OF DIMENSION 4.3. THE GAUSSIAN ALGORITHM AS A PRACTICAL METHOD FOR SOLVING LINEAR ALGEBRA PROBLEMS 4.4. EXERCISES WITH SOLUTIONS 4.5. SUGGESTED EXERCISES 4.6. APPENDIX: THE COMPLETION THEOREM Chapter 5: Linear Transformations 5.1. LINEAR TRANSFORMATIONS: DEFINITION 5.2. LINEAR MAPS AND MATRICES 5.3. THE COMPOSITION OF LINEAR TRANSFORMATIONS 5.4. KERNEL AND IMAGE 5.5. THE RANK NULLITY THEOREM 5.6. ISOMORPHISM OF VECTOR SPACES 5.7. CALCULATION OF KERNEL AND IMAGE 5.8. EXERCISES WITH SOLUTIONS 5.9. SUGGESTED EXERCISES Chapter 6: Linear Systems 6.1. PREIMAGE 6.2. LINEAR SYSTEMS 6.3. EXERCISES WITH SOLUTIONS 6.4. SUGGESTED EXERCISES Chapter 7: Determinant and Inverse 7.1. DEFINITION OF DETERMINANT 7.2. CALCULATING THE DETERMINANT: CASES 2 x 2 AND 3 x 3 7.3. CALCULATING THE DETERMINANT WITH A RECURSIVE METHOD 7.4. INVERSE OF A MATRIX 7.5. CALCULATION OF THE INVERSE WITH THE GAUSSIAN ALGORITHM 7.6. THE LINEAR MAPS FROM RN TO RN 7.7. EXERCISES WITH SOLUTIONS 7.8. SUGGESTED EXERCISES 7.9. APPENDIX Chapter 8: Change of Basis 8.1. LINEAR TRANSFORMATIONS AND MATRICES 8.2. THE IDENTITY MAP 8.3. CHANGE OF BASIS FOR LINEAR TRANSFORMATIONS 8.4. EXERCISES WITH SOLUTIONS 8.5. SUGGESTED EXERCISES Chapter 9: Eigenvalues and Eigenvectors 9.1. DIAGONALIZABILITY 9.2. EIGENVALUES AND EIGENVECTORS 9.3. EXERCISES WITH SOLUTIONS 9.4. SUGGESTED EXERCISES Chapter 10: Scalar Products 10.1. BILINEAR FORMS 10.2. BILINEAR FORMS AND MATRICES 10.3. BASIS CHANGE 10.4. SCALAR PRODUCTS 10.5. ORTHOGONAL SUBSPACES 10.6. GRAM-SCHMIDT ALGORITHM 10.7. EXERCISES WITH SOLUTIONS 10.8. SUGGESTED EXERCISES Chapter 11: Spectral Theorem 11.1. ORTHOGONAL LINEAR TRANSFORMATIONS 11.2. ORTHOGONAL MATRICES 11.3. SYMMETRIC LINEAR TRANSFORMATIONS 11.4. THE SPECTRAL THEOREM 11.5. EXERCISES WITH SOLUTIONS 11.6. SUGGESTED EXERCISES 11.7. APPENDIX: THE COMPLEX CASE Chapter 12: Applications of Spectral Theorem and Quadratic Forms 12.1. DIAGONALIZATION OF SCALAR PRODUCTS 12.2. QUADRATIC FORMS 12.3. QUADRATIC FORMS AND CURVES IN THE PLANE 12.4. EXERCISES WITH SOLUTIONS 12.5. SUGGESTED EXERCISES Chapter 13: Lines and Planes 13.1. POINTS AND VECTORS IN R3 13.2. SCALAR PRODUCT AND VECTOR PRODUCT 13.3. LINES IN R3 13.4. PLANES IN R3 13.5. EXERCISES WITH SOLUTIONS 13.6. SUGGESTED EXERCISES Chapter 14: Introduction to Modular Arithmetic 14.1. THE PRINCIPLE OF INDUCTION 14.2. THE DIVISION ALGORITHM AND EUCLID’S ALGORITHM 14.3. CONGRUENCE CLASSES 14.4. CONGRUENCES 14.5. EXERCISES WITH SOLUTIONS 14.6. SUGGESTED EXERCISES 14.7. APPENDIX: ELEMENTARY NOTIONS OF SET THEORY Appendix A: Complex Numbers A.1 COMPLEX NUMBERS A.2 POLAR REPRESENTATION Appendix B: Solutions of some suggested exercises Bibliography Index