Linear Algebra

MICHAEL L. O’LEARY, is Professor of Mathematics at College of DuPage in Glen Ellyn, Illinois. He received his doctoral degree in mathematics from the University of California, Irvine in 1994 and is the author of A First Course in Mathematical Logic and Set Theory and Revolutions of Geometry, both published by Wiley.

• Preface xiAcknowledgments xvAbout the Companion Website xvi1 Logic and Set Theory 11.1 Statements 1Connectives 2Logical Equivalence 31.2 Sets and Quantification 7Universal Quantifiers 8Existential Quantifiers 9Negating Quantifiers 10Set-Builder Notation 12Set Operations 13Families of Sets 141.3 Sets and Proofs 18Direct Proof 20Subsets 22Set Equality 23Indirect Proof 24Mathematical Induction 251.4 Functions 30Injections 33Surjections 35Bijections and Inverses 37Images and Inverse Images 40Operations 412 Euclidean Space 492.1 Vectors 49Vector Operations 51Distance and Length 57Lines and Planes 642.2 Dot Product 74Lines and Planes 77Orthogonal Projection 822.3 Cross Product 88Properties 91Areas and Volumes 933 Transformations and Matrices 993.1 Linear Transformations 99Properties 103Matrices 1063.2 Matrix Algebra 116Addition, Subtraction, and Scalar Multiplication 116Properties 119Multiplication 122Identity Matrix 129Distributive Law 132Matrices and Polynomials 1323.3 Linear Operators 137Reflections 137Rotations 142Isometries 147Contractions, Dilations, and Shears 1503.4 Injections and Surjections 155Kernel 155Range 1583.5 Gauss–Jordan Elimination 162Elementary Row Operations 164Square Matrices 167Nonsquare Matrices 171Gaussian Elimination 1774 Invertibility 1834.1 Invertible Matrices 183Elementary Matrices 186Finding the Inverse of a Matrix 192Systems of Linear Equations 1944.2 Determinants 198Multiplying a Row by a Scalar 203Adding a Multiple of a Row to Another Row 205Switching Rows 2104.3 Inverses and Determinants 215Uniqueness of the Determinant 216Equivalents to Invertibility 220Products 2224.4 Applications 227The Classical Adjoint 228Symmetric and Orthogonal Matrices 229Cramer’s Rule 234LU Factorization 236Area and Volume 2385 Abstract Vectors 2455.1 Vector Spaces 245Examples of Vector Spaces 247Linear Transformations 2535.2 Subspaces 259Examples of Subspaces 260Properties 261Spanning Sets 264Kernel and Range 2665.3 Linear Independence 272Euclidean Examples 274Abstract Vector Space Examples 2765.4 Basis and Dimension 281Basis 281Zorn’s Lemma 285Dimension 287Expansions and Reductions 2905.5 Rank and Nullity 296Rank-Nullity Theorem 297Fundamental Subspaces 302Rank and Nullity of a Matrix 3045.6 Isomorphism 310Coordinates 315Change of Basis 320Matrix of a Linear Transformation 3246 Inner Product Spaces 3356.1 Inner Products 335Norms 341Metrics 342Angles 344Orthogonal Projection 3476.2 Orthonormal Bases 352Orthogonal Complement 355Direct Sum 357Gram–Schmidt Process 361QR Factorization 3667 Matrix Theory 3737.1 Eigenvectors and Eigenvalues 373Eigenspaces 375Characteristic Polynomial 377Cayley–Hamilton Theorem 3827.2 Minimal Polynomial 386Invariant Subspaces 389Generalized Eigenvectors 391Primary Decomposition Theorem 3937.3 Similar Matrices 402Schur’s Lemma 405Block Diagonal Form 408Nilpotent Matrices 412Jordan Canonical Form 4157.4 Diagonalization 422Orthogonal Diagonalization 426Simultaneous Diagonalization 428Quadratic Forms 432Further Reading 441Index 443