Elementary Linear Algebra (eBook)

Front Cover 1
Title Page 2
Copyright Page 3
Dedication Page 4
Table of Contents 6
Preface for the Instructor 10
Preface for the Student 20
Symbol Table 24
Computational and Numerical Methods, Applications 28
Chapter 1. Vectors and Matrices 32
1.1 Fundamental Operations with Vectors 33
1.2 The Dot Product 49
1.3 An Introduction to Proof Techniques 62
1.4 Fundamental Operations with Matrices 79
1.5 Matrix Multiplication 90
Chapter 2. Systems of Linear Equations 110
2.1 Solving Linear Systems Using Gaussian Elimination 110
2.2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form 129
2.3 Equivalent Systems, Rank, and Row Space 141
2.4 Inverses of Matrices 156
Chapter 3. Determinants and Eigenvalues 174
3.1 Introduction to Determinants 174
3.2 Determinants and Row Reduction 186
3.3 Further Properties of the Determinant 196
3.4 Eigenvalues and Diagonalization 209
Chapter 4. Finite Dimensional Vector Spaces 234
4.1 Introduction to Vector Spaces 235
4.2 Subspaces 246
4.3 Span 258
4.4 Linear Independence 270
4.5 Basis and Dimension 286
4.6 Constructing Special Bases 300
4.7 Coordinatization 312
Chapter 5. Linear Transformations 336
5.1 Introduction to Linear Transformations 337
5.2 The Matrix of a Linear Transformation 352
5.3 The Dimension Theorem 369
5.4 One-to-One and Onto Linear Transformations 381
5.5 Isomorphism 387
5.6 Diagonalization of Linear Operators 402
Chapter 6. Orthogonality 428
6.1 Orthogonal Bases and the Gram-Schmidt Process 428
6.2 Orthogonal Complements 443
6.3 Orthogonal Diagonalization 459
Chapter 7. Complex Vector Spaces and General Inner Products 476
7.1 Complex n-Vectors and Matrices 477
7.2 Complex Eigenvalues and Complex Eigenvectors 485
7.3 Complex Vector Spaces 491
7.4 Orthogonality in Cn 495
7.5 Inner Product Spaces 503
Chapter 8. Additional Applications 522
8.1 Graph Theory 522
8.2 Ohm’s Law 532
8.3 Least-Squares Polynomials 535
8.4 Markov Chains 543
8.5 Hill Substitution: An Introduction to Coding Theory 556
8.6 Elementary Matrices 561
8.7 Rotation of Axes for Conic Sections 568
8.8 Computer Graphics 575
8.9 Differential Equations 592
8.10 Least-Squares Solutions for Inconsistent Systems 601
8.11 Quadratic Forms 609
Chapter 9. Numerical Methods 618
9.1 Numerical Methods for Solving Systems 619
9.2 LDU Decomposition 631
9.3 The Power Method for Finding Eigenvalues 639
9.4 QR Factorization 646
9.5 Singular Value Decomposition 654
Appendix A. Miscellaneous Proofs 676
Proof of Theorem 1.14, Part (1) 676
Proof of Theorem 2.4 677
Proof of Theorem 2.9 678
Proof of Theorem 3.3, Part (3), Case 2 679
Proof of Theorem 5.29 680
Proof of Theorem 6.18 681
Appendix B. Functions 684
Functions: Domain, Codomain, and Range 684
One-to-One and Onto Functions 685
Composition and Inverses of Functions 686
Appendix C. Complex Numbers 692
Appendix D. Answers to Selected Exercises 696
Index 756