This book is an introductory textbook in linear algebra. The authors begin with a primer on logic and set theory in the first two chapters. Functions are also covered early so that the core topic of linear transformations can be introduced near the beginning of the book and serve as a motivation to solving systems of equations. Next, Euclidean Spaces and functions between Euclidean Spaces are covered. This includes specific discussions on vectors and the standard basis, vector algebra, Pythagoras, and matrix transformations. Invertible Linear Transformations are then examined, along with Spaces without Geometry. This section includes a discussion of subspaces, linear independence, and change of basis. The authors then cover functions between spaces and geometry on vector spaces. Simplifying linear transformations is then discussed. The book concludes with two advanced chapters covering matrix theory and more general algebraic structures. 

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