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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 . . . 12

I.3 General=Particular+Homogeneous. . . 22

II Linear Geometry . . . 34

II.1 Vectors in Space* . . . 34

II.2 Length and Angle Measures* . . . 41

III Reduced Echelon Form . . . 50

III.1 Gauss-Jordan Reduction. . . 50

III.2 The Linear Combination Lemma . . . 55

Topic: Computer Algebra Systems . . . 64

Topic: Accuracy of Computations . . . 66

Topic: Analyzing Networks . . . 71

Chapter Two: Vector Spaces I Definition of Vector Space . . . 78

I.1 Definition and Examples . . . 78

I.2 Subspaces and Spanning Sets . . . 90

II Linear Independence . . . 102

II.1 Definition and Examples . . . 102

III Basis and Dimension . . . 114

III.1 Basis. . . 114

III.2 Dimension . . . 120

III.3 Vector Spaces and Linear Systems . . . 127

III.4 Combining Subspaces*. . . 134

Topic: Fields. . . 144

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Topic: Crystals . . . 146

Topic: Voting Paradoxes . . . 150

Topic: Dimensional Analysis . . . 156

Chapter Three: Maps Between Spaces I Isomorphisms . . . 165

I.1 Definition and Examples . . . 165

I.2 Dimension Characterizes Isomorphism . . . 175

II Homomorphisms . . . 183

II.1 Definition . . . 183

II.2 Range space and Null space . . . 191

III Computing Linear Maps . . . 203

III.1 Representing Linear Maps with Matrices . . . 203

III.2 Any Matrix Represents a Linear Map* . . . 213

IV Matrix Operations . . . 222

IV.1 Sums and Scalar Products. . . 222

IV.2 Matrix Multiplication . . . 226

IV.3 Mechanics of Matrix Multiplication . . . 234

IV.4 Inverses . . . 244

V Change of Basis . . . 252

V.1 Changing Representations of Vectors . . . 252

V.2 Changing Map Representations . . . 256

VI Projection . . . 264

VI.1 Orthogonal Projection Into a Line* . . . 264

VI.2 Gram-Schmidt Orthogonalization* . . . 269

VI.3 Projection Into a Subspace*. . . 274

Topic: Line of Best Fit . . . 283

Topic: Geometry of Linear Maps . . . 289

Topic: Magic Squares . . . 296

Topic: Markov Chains . . . 302

Topic: Orthonormal Matrices . . . 308

Chapter Four: Determinants I Definition . . . 316

I.1 Exploration* . . . 316

I.2 Properties of Determinants . . . 321

I.3 The Permutation Expansion . . . 326

I.4 Determinants Exist* . . . 335

II Geometry of Determinants . . . 344

II.1 Determinants as Size Functions . . . 344

III Laplace’s Formula. . . 351

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III.1 Laplace’s Expansion* . . . 351

Topic: Cramer’s Rule . . . 357

Topic: Speed of Calculating Determinants . . . 360

Topic: Chiò’s Method. . . 363

Topic: Projective Geometry . . . 368

Chapter Five: Similarity I Complex Vector Spaces . . . 381

I.1 Polynomial Factoring and Complex Numbers* . . . 382

I.2 Complex Representations . . . 384

II Similarity . . . 386

II.1 Definition and Examples . . . 386

II.2 Diagonalizability . . . 389

II.3 Eigenvalues and Eigenvectors . . . 393

III Nilpotence . . . 403

III.1 Self-Composition* . . . 403

III.2 Strings* . . . 407

IV Jordan Form. . . 419

IV.1 Polynomials of Maps and Matrices* . . . 419

IV.2 Jordan Canonical Form*. . . 427

Topic: Method of Powers . . . 442

Topic: Stable Populations . . . 446

Topic: Page Ranking . . . 448

Topic: Linear Recurrences . . . 452

Appendix

Statements . . . A-1 Quantifiers . . . A-2 Techniques of Proof . . . A-3 Sets, Functions, and Relations. . . A-5

∗Starred subsections are optional.

Referensi

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