Linear Algebra Fourth Edition PDF Free Download

Linear Algebra Fourth Edition by Jim Hefferon

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Contents

Chapter One: Linear Systems

I Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 1

I.1 Gauss’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 2

I.2 Describing the Solution Set . . . . . . . . . . . . . . . . . . . 13

I.3 General = Particular + Homogeneous . . . . . . . . . . . . . . 23

II Linear Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

II.1 Vectors in Space* . . . . . . . . . . . . . . . . . . . . . . . . 35

II.2 Length and Angle Measures* . . . . . . . . . . . . . . . . . . 42

III Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . 50

III.1 Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . 50

III.2 The Linear Combination Lemma . . . . . . . . . . . . . . . . 56

Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . 65

Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . 67

Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . 72

Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter Two: Vector Spaces

I Definition of Vector Space . . . . . . . . . . . . . . . . . . . . . . 84

I.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . 84

I.2 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . 96

II Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 108

II.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . 108

III Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 121

III.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

III.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

III.3 Vector Spaces and Linear Systems . . . . . . . . . . . . . . . 136

III.4 Combining Subspaces* . . . . . . . . . . . . . . . . . . . . . . 144

Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Topic: Voting Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . 159

Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . 165

Chapter Three: Maps Between Spaces

I Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

I.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . 173

I.2 Dimension Characterizes Isomorphism . . . . . . . . . . . . . 183

II Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

II.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

II.2 Range Space and Null Space . . . . . . . . . . . . . . . . . . 199

III Computing Linear Maps . . . . . . . . . . . . . . . . . . . . . . . 212

III.1 Representing Linear Maps with Matrices . . . . . . . . . . . 212

III.2 Any Matrix Represents a Linear Map . . . . . . . . . . . . . 223

IV Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 232

IV.1 Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . 232

IV.2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . 236

IV.3 Mechanics of Matrix Multiplication . . . . . . . . . . . . . . 244

IV.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

V Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

V.1 Changing Representations of Vectors . . . . . . . . . . . . . . 262

V.2 Changing Map Representations . . . . . . . . . . . . . . . . . 267

VI Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

VI.1 Orthogonal Projection Into a Line* . . . . . . . . . . . . . . 275

VI.2 Gram-Schmidt Orthogonalization* . . . . . . . . . . . . . . . 280

VI.3 Projection Into a Subspace* . . . . . . . . . . . . . . . . . . . 285

Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . . 301

Topic: Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . 319

Chapter Four: Determinants

I Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

I.1 Exploration* . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

I.2 Properties of Determinants . . . . . . . . . . . . . . . . . . . 331

I.3 The Permutation Expansion . . . . . . . . . . . . . . . . . . 337

I.4 Determinants Exist* . . . . . . . . . . . . . . . . . . . . . . . 346

II Geometry of Determinants . . . . . . . . . . . . . . . . . . . . . . 355

II.1 Determinants as Size Functions . . . . . . . . . . . . . . . . . 355

III Laplace’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

III.1 Laplace’s Expansion* . . . . . . . . . . . . . . . . . . . . . . 363

Topic: Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Topic: Speed of Calculating Determinants . . . . . . . . . . . . . . . 372

Topic: Chiò’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

Topic: Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . 380

Topic: Computer Graphics . . . . . . . . . . . . . . . . . . . . . . . . 392

Chapter Five: Similarity

I Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 397

I.1 Polynomial Factoring and Complex Numbers* . . . . . . . . 398

I.2 Complex Representations . . . . . . . . . . . . . . . . . . . . 400

II Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

II.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . 402

II.2 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . 407

II.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 412

III Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

III.1 Self-Composition* . . . . . . . . . . . . . . . . . . . . . . . . 424

III.2 Strings* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

IV Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

IV.1 Polynomials of Maps and Matrices* . . . . . . . . . . . . . . 440

IV.2 Jordan Canonical Form* . . . . . . . . . . . . . . . . . . . . . 448

Topic: Method of Powers . . . . . . . . . . . . . . . . . . . . . . . . . 464

Topic: Stable Populations . . . . . . . . . . . . . . . . . . . . . . . . 468

Topic: Page Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

Topic: Linear Recurrences . . . . . . . . . . . . . . . . . . . . . . . . 474

Topic: Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . . . . 482

Appendix

Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Techniques of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

Sets, Functions, and Relations . . . . . . . . . . . . . . . . . . . . . A-5

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