510 INDEX

connected component, 260connected components, 260

equivalence class, 260equivalence relation, 260open sets, 260

connected setsintersection, 259intervals, 260real line, 260

connecting sequence, 475consistent, 44continuous function, 251continuous functions, 284

equivalent conditions, 251contraction map, 255

fixed point theorem, 255convex combination, 116, 280convex hull, 116, 280

compactness, 116convex set

homology, 461Coordinates, 53countable, 4counting zeros, 336Courant Fischer theorem, 369cow lick, 493Cramer’s rule, 181cyclic basis, 148cyclic set, 147

De Moivre’s theorem, 12degree

antipodal map, 486Euclidean space, 491odd function, 501on spheres, 484orientation, 496product formula, 491properties, 492regular value, 500switching variables, 485well defined, 488

derivative, 284determinant

definition, 174estimate for Hermitian matrix, 388expansion along row, column, 179Hadamard inequality, 388

matrix inverse, 179partial derivative, cofactor, 190permutation of rows, 174product, 176product of eigenvalues, 341row, column operations, 175summary of properties, 185symmetric definition, 175transpose, 175

diagonal matrix, 135diagonalizability, 135diagonalizable, 135, 357

formal derivative, 140minimal polynomial and its derivative,

140differential equations

first order systems, 343dimension of a vector space, 262dimension of vector space, 54direct sum, 119, 158, 460

minimum polynomial splits, 162notation, 119

discrete Fourier transform, 384distance, 245

to a subspace, 313distance to a nonempty set, 247distinct eigenvalues, 147distinct roots

polynomial and its derivative, 239dot product, 263dyadics, 103

echelon form, 41eigen-pair, 132eigenvalue, 132

existence, 133eigenvalues, 186, 336

AB and BA, 187eigenvector, 132

existence, 133eigenvectors

distinct eigenvalues, 147independent, 147

elementary matrices, 91product, 95

elementary matrixinverse, 95properties, 95