INDEX 511

elementary operations, 39elementary symmetric polynomials, 196empty set, 1equivalence class, 6, 63, 107

of polynomials, 64equivalence relation, 6, 63, 107Euclidean algorithm, 19exchange theorem, 54existence of a fixed point, 401

factorization of matrixEuclidean domain, 151

field axioms, 8field extension

dimension, 67finite, 67

field extensions, 67Field of scalars, 51fields

characteristic, 240perfect, 241

fieldsperfect, 240

finite dimensional vector space, 54finite fields, 27fixed field, 219fixed fields and subgroups, 224fixed point property, 483formal derivative, 140Fourier series, 323Fredholm alternative, 318Frobenius

inner product, 347Frobenius norm, 275, 374

singular value decomposition, 374Frobinius norm, 383function, 3fundamental theorem of algebra, 13, 15, 210fundamental theorem of algebra

plausibility argument, 15rigorous proof, 16

fundamental theorem of arithmetic, 20fundamental theorem of Galois theory, 226

Galois groupsize, 217

Gauss Elimination, 44Gauss elimination, 40

Gauss Jordan method for inverses, 82Gauss Seidel method, 402geometric simplices

boundary, 466Gerschgorin’s theorem, 334Gram Schmidt process, 271, 309Grammian matrix, 312, 319

invertible, 312greatest common divisor, 19, 22

characterization, 19description, 22

Gronwall’s inequality, 407group

definition, 217group

solvable, 232

Hadamardinequality, 388

Hamel basis, 505Hausdorff

maximal principle, 503Hermitian, 331

orthonormal basis eigenvectors, 367positive definite, 371

Hermitian matrixfactorization, 387positive part, 382positive part, Lipschitz continuous, 382

Hermitian operator, 315largest, smallest, eigenvalues, 368

Hilbert space, 307Holder’s inequality, 268homeomorphism, 455homogeneous coordinates, 115homology

convex spaces, 461of spheres, 480path components, 460pathconnected, 459spheres, 480

homology group, 458homomorphism, 211, 456

boundary, 456connecting, 476

homotopic, 463maps, 465

homotopy