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 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


Re_ections 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