Preface xi
Acknowledgments xv
About the Companion Website xvi
1 Logic and Set Theory 1
1.1 Statements 1
Connectives 2
Logical Equivalence 3
1.2 Sets and Quantification 7
Universal Quantifiers 8
Existential Quantifiers 9
Negating Quantifiers 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 Gaussâ"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
Cramerâs 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
Zornâs 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
Gramâ"Schmidt Process 361
QR Factorization 366
7 Matrix Theory 373
7.1 Eigenvectors and Eigenvalues 373
Eigenspaces 375
Characteristic Polynomial 377
Cayleyâ"Hamilton Theorem 382
7.2 Minimal Polynomial 386
Invariant Subspaces 389
Generalized Eigenvectors 391
Primary Decomposition Theorem 393
7.3 Similar Matrices 402
Schurâs 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
