Description
Preface xi Acknowledgments xv 1 Logic and Set Theory 1 1.1 Statements 1 Connectives 2 Logical Equivalence 3 1.2 Sets and Quantification 7 Universal Quantification 8 Existential Quantification 9 Negating Quantification 10 Set-Builder Notation 12 Set Operations 13 Families of Sets 14 1.3 Sets and Proofs 18 Direct Proof 20 Subsets 22 Set Equality 23 Indirect Proof 24 Mathematical Induction 25 1.4 Functions 30 Injections 33 Surjections 35 Bijections and Inverses 37 Images and Inverse Images 40 Operations 41 2 Euclidean Space 49 2.1 Vectors 49 Vector Operations 51 Distance and Length 57 Lines and Planes 64 2.2 Dot Product 74 Lines and Planes 77 Orthogonal Projection 82 2.3 Cross Product 88 Properties 91 Areas and Volumes 93 3 Transformations and Matrices 99 3.1 Linear Transformations 99 Properties 103 Matrices 106 3.2 Matrix Algebra 116 Addition, Subtraction, and Scalar Multiplication 116 Properties 119 Multiplication 122 Identity Matrix 129 Distributive Law 132 Matrices and Polynomials 132 3.3 Linear Operators 137 Reflections 137 Rotations 142 Isometries 147 Contractions, Dilations, and Shears 150 3.4 Injections and Surjections 155 Kernel 155 Range 158 3.5 Gauss-Jordan Elimination 162 Elementary Row Operations 164 Square Matrices 167 Nonsquare Matrices 171 Gaussian Elimination 177 4 Invertibility 183 4.1 Invertible Matrices 183 Elementary Matrices 186 Finding the Inverse of a Matrix 192 Systems of Linear Equations 194 4.2 Determinants 198 Multiplying a Row by a Scalar 203 Adding a Multiple of a Row to Another Row 205 Switching Rows 210 4.3 Inverses and Determinants 215 Uniqueness of the Determinant 216 Equivalents to Invertibility 220 Products 222 4.4 Applications 227 The Classical Adjoint 228 Symmetric and Orthogonal Matrices 229 Cramer’s Rule 234 LU Factorization 236 Area and Volume 238 5 Abstract Vectors 245 5.1 Vector Spaces 245 Examples of Vector Spaces 247 Linear Transformations 253 5.2 Subspaces 259 Examples of Subspaces 260 Properties 261 Spanning Sets 264 Kernel and Range 266 5.3 Linear Independence 272 Euclidean Examples 274 Abstract Vector Space Examples 276 5.4 Basis and Dimension 281 Basis 281 Zorn’s Lemma 285 Dimension 287 Expansions and Reductions 290 5.5 Rank and Nullity 296 Rank-Nullity Theorem 297 Fundamental Subspaces 302 Rank and Nullity of a Matrix 304 5.6 Isomorphism 310 Coordinates 315 Change of Basis 320 Matrix of a Linear Transformation 324 6 Inner Product Spaces 335 6.1 Inner Products 335 Norms 341 Metrics 342 Angles 344 Orthogonal Projection 347 6.2 Orthonormal Bases 352 Orthogonal Complement 355 Direct Sum 357 Gram-Schmidt Process 361 QR Factorization 366 7 Matrix Theory 373 7.1 Eigenvectors and Eigenvalues 373 Eigenspaces 375 Characteristic Polynomial 377 Cayley-Hamilton Theorem 382 7.2 Minimal Polynomial 386 Invariant Subspaces 389 Generalized Eigenvectors 391 Primary Decomposition Theorem 393 7.3 Similar Matrices 402 Schur’s Lemma 405 Block Diagonal Form 408 Nilpotent Matrices 412 Jordan Canonical Form 415 7.4 Diagonalization 422 Orthogonal Diagonalization 426 Simultaneous Diagonalization 428 Quadratic Forms 432 Further Reading 441 Index 443
