INDICE
CITA
Contents
Chapter One: Linear Systems 3
Subsection One.I.1: Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Subsection One.I.2: Describing the Solution Set . . . . . . . . . . . . . . . . . . . . . . . 10
Subsection One.I.3: General = Particular + Homogeneous . . . . . . . . . . . . . . . . . 14
Subsection One.II.1: Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Subsection One.II.2: Length and Angle Measures . . . . . . . . . . . . . . . . . . . . . . 20
Subsection One.III.1: Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . . . 25
Subsection One.III.2: Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter Two: Vector Spaces 36
Subsection Two.I.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 37
Subsection Two.I.2: Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . . . . 40
Subsection Two.II.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 46
Subsection Two.III.1: Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Subsection Two.III.2: Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Subsection Two.III.3: Vector Spaces and Linear Systems . . . . . . . . . . . . . . . . . . 61
Subsection Two.III.4: Combining Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 66
Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Topic: Voting Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter Three: Maps Between Spaces 74
Subsection Three.I.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . 75
Subsection Three.I.2: Dimension Characterizes Isomorphism . . . . . . . . . . . . . . . . 83
Subsection Three.II.1: Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Subsection Three.II.2: Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . . 90
Subsection Three.III.1: Representing Linear Maps with Matrices . . . . . . . . . . . . . 95
Subsection Three.III.2: Any Matrix Represents a Linear Map . . . . . . . . . . . . . . . 103
Subsection Three.IV.1: Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . 107
Subsection Three.IV.2: Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 108
Subsection Three.IV.3: Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . 112
Subsection Three.IV.4: Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Subsection Three.V.1: Changing Representations of Vectors . . . . . . . . . . . . . . . . 121
Subsection Three.V.2: Changing Map Representations . . . . . . . . . . . . . . . . . . . 124
Subsection Three.VI.1: Orthogonal Projection Into a Line . . . . . . . . . . . . . . . . . 128
Subsection Three.VI.2: Gram-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . 131
Subsection Three.VI.3: Projection Into a Subspace . . . . . . . . . . . . . . . . . . . . . 137
Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Chapter Four: Determinants 158
Subsection Four.I.1: Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Subsection Four.I.2: Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . 161
Subsection Four.I.3: The Permutation Expansion . . . . . . . . . . . . . . . . . . . . . . 164
Subsection Four.I.4: Determinants Exist . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Subsection Four.II.1: Determinants as Size Functions . . . . . . . . . . . . . . . . . . . . 168
Subsection Four.III.1: Laplace’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 171
Topic: Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Topic: Speed of Calculating Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Topic: Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Chapter Five: Similarity 178
Subsection Five.II.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 179
Subsection Five.II.2: Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Subsection Five.II.3: Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . 186
Subsection Five.III.1: Self-Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Subsection Five.III.2: Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Subsection Five.IV.1: Polynomials of Maps and Matrices . . . . . . . . . . . . . . . . . . 196
Subsection Five.IV.2: Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . 203
Topic: Method of Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Topic: Stable Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Topic: Linear Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Linear Algebra - Answers to Exercises, Jim Hefferon
Jun 12 2007, 06:18 PM
