INDEX 479

row operations, 112transpose, 133zero, 176

determinant rankrow rank, 176

diagonal matrix, 261, 284diagonalizable, 261, 284, 297, 443differential equations

first order systems, 400dimension, 167

definition, 168dimension of vector space, 380direct sum, 428distance formula, 20Dolittle’s method, 207dot product, 31

properties, 31duality, 240dynamical system, 285

echelon form, 58, 59eigenspace, 250eigenvalue, 248, 432

existence, 430eigenvalues, 139eigenvector, 248Einstein summation convention, 48elementary matrices, 143elementary matrix

inverse, 146properties, 146

elementary operations, 55empty set, 1entries of a matrix, 82equivalence class, 388, 442equivalence relation, 387, 442exchange theorem, 165, 377

field axioms, 4field extension, 388

dimension, 390finite, 390

field extensions, 391Field of scalars, 373force, 25Fourier coefficients, 411Fredholm alternative, 174, 313free variables, 65, 222

Frobinius norm, 330fundamental matrix, 448fundamental theorem of algebra, 9, 10fundamental theorem of algebra

plausibility argument, 11rigorous proof, 12

Gauss Elimination, 67Gauss elimination, 57, 58Gauss Jordan method for inverses, 95Gauss Seidel, 335Gauss Seidel method, 335general solution, 197

solution space, 195generalized eigenspace, 432

direct sum, 429geometric multiplicity, 255Gerschgorin’s theorem, 278Gram Schmidt process, 305, 407Grammian matrix, 408greatest common divisor, 384

Hermitian, 308homogeneous coordinates, 199homogeneous syster, 195homomorphism, 185Householder matrix, 213householder matrix, 199

inconsistent, 64, 67independent set

extending to a basis, 169independent set of vectors

extending to form a basis, 169injective, 425inner produc

strange examplet, 40inner product, 31, 37

axioms, 403Cauchy Schwarz inequality, 404

inner productproperties, 38

integers mod a prime, 376intersection, 1intervals

notation, 1inverse

left inverse, 138right inverse, 138