Linear Algebra. Answers to Exercises

Chapter One: Linear Systems 7 Subsection One.I.1: Gauss' Method 9 Subsection One.I.2: Describing the Solution Set 10 Subsection One.I.3: General = Particular + Homogeneous 17 Subsection One.II.1: Vectors in Space 21 Subsection One.II.2: Length and Angle Measures 23 Subsection One.III.1: Gauss-Jordan Reduction 25 Subsection One.III.2: Row Equivalence 31 Topic: Computer Algebra Systems 35 Topic: Input-Output Analysis 36 Topic: Accuracy of Computations 37 Topic: Analyzing Networks 38 Chapter Two: Vector Spaces 40 Subsection Two.I.1: Definition and Examples 41 Subsection Two.I.2: Subspaces and Spanning Sets 44 Subsection Two.II.1: Definition and Examples 50 Subsection Two.III.1: Basis 57 Subsection Two.III.2: Dimension 61 Subsection Two.III.3: Vector Spaces and Linear Systems 64 Subsection Two.III.4: Combining Subspaces 69 Topic: Fields 73 Topic: Crystals 73 Topic: Dimensional Analysis 75 Chapter Three: Maps Between Spaces 77 Subsection Three.I.1: Definition and Examples 79 Subsection Three.I.2: Dimension Characterizes Isomorphism 87 Subsection Three.II.1: Definition 89 Subsection Three.II.2: Rangespace and Nullspace 94 Subsection Three.III.1: Representing Linear Maps with Matrices 98 Subsection Three.III.2: Any Matrix Represents a Linear Map 107 Subsection Three.IV.1: Sums and Scalar Products 111 Subsection Three.IV.2: Matrix Multiplication 112 Subsection Three.IV.3: Mechanics of Matrix Multiplication 116 Subsection Three.IV.4: Inverses 120 Subsection Three.V.1: Changing Representations of Vectors 125 Subsection Three.V.2: Changing Map Representations 129 Subsection Three.VI.1: Orthogonal Projection Into a Line 132 Subsection Three.VI.2: Gram-Schmidt Orthogonalization 135 Subsection Three.VI.3: Projection Into a Subspace 141 Topic: Line of Best Fit 148 Topic: Geometry of Linear Maps 152 Topic: Markov Chains 155 Topic: Orthonormal Matrices 162 Chapter Four: Determinants 163 Subsection Four.I.1: Exploration 165 Subsection Four.I.2: Properties of Determinants 167 Subsection Four.I.3: The Permutation Expansion 169 Subsection Four.I.4: Determinants Exist 171 Subsection Four.II.1: Determinants as Size Functions 174 Subsection Four.III.1: Laplace's Expansion 177 Topic: Cramer's Rule 180 Topic: Speed of Calculating Determinants 180 Topic: Projective Geometry 182 Chapter Five: Similarity 184 Subsection Five.II.1: Definition and Examples 185 Subsection Five.II.2: Diagonalizability 185 Subsection Five.II.3: Eigenvalues and Eigenvectors 191 Subsection Five.III.1: Self-Composition 196 Subsection Five.III.2: Strings 197 Subsection Five.IV.1: Polynomials of Maps and Matrices 201 Subsection Five.IV.2: Jordan Canonical Form 208 Topic: Method of Powers 215 Topic: Stable Populations 216 Topic: Linear Recurrences 216 Chapter One: Linear Systems 217 Subsection One.I.1: Gauss' Method 219 Subsection One.I.2: Describing the Solution Set 220 Subsection One.I.3: General = Particular + Homogeneous 227 Subsection One.II.1: Vectors in Space 231 Subsection One.II.2: Length and Angle Measures 233 Subsection One.III.1: Gauss-Jordan Reduction 235 Subsection One.III.2: Row Equivalence 241 Topic: Computer Algebra Systems 245 Topic: Input-Output Analysis 246 Topic: Accuracy of Computations 247 Topic: Analyzing Networks 248 Chapter Two: Vector Spaces 250 Subsection Two.I.1: Definition and Examples 251 Subsection Two.I.2: Subspaces and Spanning Sets 254 Subsection Two.II.1: Definition and Examples 260 Subsection Two.III.1: Basis 267 Subsection Two.III.2: Dimension 271 Subsection Two.III.3: Vector Spaces and Linear Systems 274 Subsection Two.III.4: Combining Subspaces 279 Topic: Fields 283 Topic: Crystals 283 Topic: Dimensional Analysis 285 Chapter Three: Maps Between Spaces 287 Subsection Three.I.1: Definition and Examples 289 Subsection Three.I.2: Dimension Characterizes Isomorphism 297 Subsection Three.II.1: Definition 299 Subsection Three.II.2: Rangespace and Nullspace 304 Subsection Three.III.1: Representing Linear Maps with Matrices 308 Subsection Three.III.2: Any Matrix Represents a Linear Map 317 Subsection Three.IV.1: Sums and Scalar Products 321 Subsection Three.IV.2: Matrix Multiplication 322 Subsection Three.IV.3: Mechanics of Matrix Multiplication 326 Subsection Three.IV.4: Inverses 330 Subsection Three.V.1: Changing Representations of Vectors 335 Subsection Three.V.2: Changing Map Representations 339 Subsection Three.VI.1: Orthogonal Projection Into a Line 342 Subsection Three.VI.2: Gram-Schmidt Orthogonalization 345 Subsection Three.VI.3: Projection Into a Subspace 351 Topic: Line of Best Fit 358 Topic: Geometry of Linear Maps 362 Topic: Markov Chains 365 Topic: Orthonormal Matrices 372 Chapter Four: Determinants 373 Subsection Four.I.1: Exploration 375 Subsection Four.I.2: Properties of Determinants 377 Subsection Four.I.3: The Permutation Expansion 379 Subsection Four.I.4: Determinants Exist 381 Subsection Four.II.1: Determinants as Size Functions 384 Subsection Four.III.1: Laplace's Expansion 387 Topic: Cramer's Rule 390 Topic: Speed of Calculating Determinants 390 Topic: Projective Geometry 392 Chapter Five: Similarity 394 Subsection Five.II.1: Definition and Examples 395 Subsection Five.II.2: Diagonalizability 395 Subsection Five.II.3: Eigenvalues and Eigenvectors 401 Subsection Five.III.1: Self-Composition 406 Subsection Five.III.2: Strings 407 Subsection Five.IV.1: Polynomials of Maps and Matrices 411 Subsection Five.IV.2: Jordan Canonical Form 418 Topic: Method of Powers 425 Topic: Stable Populations 426 Topic: Linear Recurrences 426 linear algebra,mathematics,textbook linear algebra