Linear Algebra

1 Systems of Linear Equations 1 1.1 The Vector Space of m × n Matrices 1 The Space R n 4 Linear Combinations and Linear Dependence 6 What Is a Vector Space? 10 Exercises 16 1.1.1 Computer Projects/Exercises/Exercises 21 Introduction to MATLAB 21 1.1.2 Applications to Graph Theory I 24 Exercises 26 1.2 Systems 27 Rank: The Maximum Number of Linearly Independent Equations 33 Exercises 37 1.2.1 Computer Projects/Exercises 39 The Translation Theorem 39 1.2.2 Applications to Circuit Theory 40 Exercises 44 1.3 Gaussian Elimination 45 Spanning in Polynomial Spaces 56 Computational Issues: Pivoting 59 Exercises 60 Computational Issues: Flops 65 1.3.1 Computer Projects/Exercises 66 Using tolerances in rref and rank 66 1.3.2 Applications to Traffic Flow 69 Exercises 70 1.4 Column Space and Nullspace 71 Subspaces 73 Exercises 82 1.4.1 Computer Projects/Exercises 90 The null Command 90 Chapter Summary 92 2 Linear Independence and Dimension 93 2.1 The Test for Linear Independence 93 Bases for the Column Space 99 Testing Functions for Independence 102 Exercises 104 2.1.1 Computer Projects/Exercises 108 Changing Pivot Columns 108 2.2 Dimension 109 Exercises 118 2.2.1 Computer Projects/Exercises 124 2.2.2 Applications to Differential Equations 125 Exercises 128 2.3 Row Space and the rank-nullity theorem 128 Bases for the Row Space 130 Computational Issues: Computing Rank 138 Exercises 140 2.3.1 Computer Projects/Exercises 144 Random Matrices of a given Rank 144 Chapter Summary 145 3 Linear Transformations 147 3.1 The Linearity Properties 147 Exercises 154 3.1.1 Computer Projects/Exercises 159 2-D Computer Graphics 159 3.2 Matrix Multiplication (Composition) 161 Partitioned Matrices 168 Computational Issues: Parallel Computing 170 Exercises 171 3.2.1 Computer Projects/Exercises 176 3-D Computer Graphics 176 3.2.2 Applications to Graph Theory II 177 Exercises 179 3.2.3 Computer Projects/Exercises 179 Google's Page Rank Algorithim 179 Exercises 182 3.3 Inverses 183 Computational Issues: Reduction versus Inverses 189 Exercises 191 3.3.1 Computer Projects/Exercises 196 Ill-Conditioned Systems 196 3.3.2 Applications to Economics: The Leontief open model 198 Exercises 203 3.4 The LU Factorization 204 Exercises 213 3.4.1 Computer Projects/Exercises 215 Row Exchanges in the LU Factorization 215 3.5 The Matrix of a Linear Transformation 216 Coordinates 216 Isomorphism 227 Invertible Linear Transformations 228 Exercises 230 3.5.1 Computer Projects/Exercises 234 Graphing in Skewed-Coordinates 234 3.5.2 Computer Projects/Exercises 236 Pricing Long Term Health Care Insurance 236 Exercises 240 Chapter Summary 241 4 Determinants 243 4.1 Definition of the Determinant 243 4.1.1 The Rest of the Proofs 251 Exercises 254 4.1.2 Computer Projects/Exercises 257 4.2 Reduction and Determinants 257 Uniqueness of the Determinant 262 Exercises 265 4.2.1 Volume 267 Exercises 270 4.3 A Formula for Inverses 270 Cramer's Rule 272 Exercises 275 Chapter Summary 276 5 Eigenvectors and Eigenvalues 279 5.1 Eigenvectors 279 Exercises 288 5.1.1 Computer Projects/Exercises 291 Computing Roots of Polynomials 291 5.1.2 Application to Markov Chains 292 Application to the Auto Rental Business 292 Exercises 294 5.2 Diagonalization 296 Powers of Matrices 298 Exercises 299 5.2.1 Application to Systems of Differential Equations 301 Exercises 304 5.3 Complex Eigenvectors 304 Complex Vector Spaces 311 Exercises 312 5.3.1 Computer Projects/Exercises 314 Complex Eigenvalues 314 Exercises 314 Chapter Summary 314 6 Orthogonality 317 6.1 The Scalar Product in R n 317 Orthogonal/Orthonormal Bases and Coordinates 321 Exercises 325 6.2 Projections: The Gram-Schmidt Process 327 The QR Decomposition 333 Uniqueness of the QR-factorization 336 Exercises 337 6.2.1 Computer Projects/Exercises 340 The Least Squares Solution 340 6.3 Fourier Series: Scalar Product Spaces 342 Exercises 348 6.3.1 Computer Projects/Exercises 352 Plotting Fourier Series 352 6.4 Orthogonal Matrices 353 Householder Matrices 359 Exercises 363 Discrete Wavelet Transform 366 6.4.1 Computer Projects/Exercises 367 6.5 Least Squares 369 Exercises 375 6.5.1 Computer Projects/Exercises 379 Finding the Orbit of an Asteroid 379 6.6 Quadratic Forms: Orthogonal Diagonalization 380 The Spectral Theorem 383 The Principal Axis Theorem 384 Exercises 390 6.6.1 Computer Projects/Exercises 393 The Principal Axis Theorem 393 6.7 The Singular Value Decomposition (SVD) 394 Application of the SVD to Least-Squares Problems 400 Exercises 402 Computing the SVD Using Householder Matrices 404 Diagonalizing Symmetric Matrices 406 6.8 Hermitian Symmetric and Unitary Matrices 408 Exercises 414 Chapter Summary 417 7 Generalized Eigenvectors 419 7.1 Generalized Eigenvectors 419 Exercises 427 7.2 Chain Bases 430 Jordan Form 436 Exercises 442 The Cayley-Hamilton Theorem 443 Chapter Summary 444 8 Numerical Techniques 445 8.1 Condition Number 445 Norms 445 Condition Number 447 Least Squares 450 Exercises 450 8.2 Computing Eigenvalues 451 Iteration 451 The QR Method 455 Exercises 461 Chapter Summary 462 Answers and Hints 464 Index 489