Index∩, 1∪, 1ε net, 253

A close to Beigenvalues, 336

Abel’s formula, 189, 194Abelian group, 51

free, 455absolute convergence

convergence, 394absolute value

complex number, 10accumulation point, 246adjoint, 329

of matrix, 275adjugate, 180affine maps, 470algebraic number

minimum polynomial, 70algebraic numbers, 69

field, 71alternating group, 228

3 cycles, 229analytic function of matrix, 302arcwise connected, 261ascending chains, 153at most countable, 4automorphism, 211axiom of choice, 4

barycenter, 469basis, 262

existence, 55basis of eigenvectors

diagonalizable, 136basis of vector space, 54Bernstein polynomial

approximation of derivative, 444Binet Cauchy

volumes, 326Binet Cauchy formula, 177binomial theorem, 32block diagonal matrices

direct sum of subspaces, 120block diagonal matrix, 120block matrix, 88block multiplication, 88, 89Borsuk Ulam theorem, 497boundary operator, 457

bounded linear transformations, 274Brouwer

fixed point theorem, 482, 483, 493

Cauchy interlacing theorem, 348, 350Cauchy Schwarz inequality, 264Cauchy sequence, 248, 324Cayley Hamilton theorem, 186, 191, 192,

384chain, 503chain complex, 478chain homomorphisms, 478chain homotopy, 465characteristic polynomial, 185Cholesky factorization, 387closed set, 246closed sets

limit points, 246closure of a set, 250cofactor, 178column rank, 183commutative ring, 67, 195

Noetherian, 169commutative ring with unity, 28commutator, 232, 348commutator subgroup, 232compact

sequentially compact, 283compact set, 253compactness

closed interval, 249equivalent conditions, 253

companion matrix, 419complete, 399complex conjugate, 10complex numbers, 8complex numbers

arithmetic, 8roots, 12triangle inequality, 10

components of a vector, 262composition of linear transformations, 111condition number, 390cone, 468conjugate

of a product, 33conjugate fields, 237conjugate linear, 314connected, 259

509