INTRODUCTION TO LINEAR ALGEBRA, 3rd Edition

  TABLE OF CONTENTS

1 Introduction to Vectors 				  1

  1.1 Vectors and Linear Combinations 			  1
  1.2 Lengths and Dot Products 				 10

2 Solving Linear Equations 				 21

  2.1 Vectors and Linear Equations 			 21
  2.2 The Idea of Elimination 				 35
  2.3 Elimination Using Matrices 			 46
  2.4 Rules for Matrix Operations 			 56
  2.5 Inverse Matrices 					 71
  2.6 Elimination = Factorization: A = LU 		 83
  2.7 Transposes and Permutations 			 96

3 Vector Spaces and Subspaces 				111

  3.1 Spaces of Vectors 				111
  3.2 The Nullspace of A: Solving Ax = 0 		122
  3.3 The Rank and the Row Reduced Form 		134
  3.4 The Complete Solution to Ax = b 			144
  3.5 Independence, Basis and Dimension 		157
  3.6 Dimensions of the Four Subspaces 			173

4 Orthogonality 					184

  4.1 Orthogonality of the Four Subspaces 		184
  4.2 Projections 					194
  4.3 Least Squares Approximations 			206
  4.4 Orthogonal Bases and Gram-Schmidt 		219

5 Determinants 						233

  5.1 The Properties of Determinants 			233
  5.2 Permutations and Cofactors 			245
  5.3 Cramer's Rule, Inverses, and Volumes 		259

6 Eigenvalues and Eigenvectors 				274

  6.1 Introduction to Eigenvalues 			274
  6.2 Diagonalizing a Matrix 				288
  6.3 Applications to Differential Equations 		304
  6.4 Symmetric Matrices 				318
  6.5 Positive Definite Matrices 			330
  6.6 Similar Matrices 					343
  6.7 Singular Value Decomposition (SVD) 		352

7 Linear Transformations 				363

  7.1 The Idea of a Linear Transformation 		363
  7.2 The Matrix of a Linear Transformation 		371
  7.3 Change of Basis 					384
  7.4 Diagonalization and the Pseudoinverse 		391

8 Applications 						401

  8.1 Matrices in Engineering 				401
  8.2 Graphs and Networks 				412
  8.3 Markov Matrices and Economic Models 		423
  8.4 Linear Programming 				431
  8.5 Fourier Series: Linear Algebra for Functions 	437
  8.6 Computer Graphics 				444

9 Numerical Linear Algebra 				450

  9.1 Gaussian Elimination in Practice 			450
  9.2 Norms and Condition Numbers 			459
  9.3 Iterative Methods for Linear Algebra 		466

10 Complex Vectors and Matrices 			477

  10.1 Complex Numbers 					477
  10.2 Hermitian and Unitary Matrices 			486
  10.3 The Fast Fourier Transform 			495

Solutions to Selected Exercises 			502
A Final Exam 						542
Matrix Factorizations 					544
Conceptual Questions for Review 			546
Glossary: A Dictionary for Linear Algebra 		551
Index 							559
Teaching Codes 						567