Index∩, 1∪, 1σ(A), 432

Abel’s formula, 127Abelian group, 373absolute value

complex number, 5adjoint, 306, 311adjugate, 115, 138algebraic multiplicity, 255algebraic number

minimal polynomial, 393algebraic numbers, 393

field, 395angle between vectors, 33area

parallelogram, 44area of a parallelogram, 42augmented matrix, 57axioms for a norm, 39

back substitution, 56bases, 166basic feasible solution, 222

finding one, 238basic variables, 65, 222basis, 166, 376

any two same size, 380column space, 168row space, 168

basis of eigenvectorsdiagonalizable, 262

bijective, 425binomial theorem, 12, 13block matrix, 301block multiplication, 300box product, 47

Cartesian coordinates, 16Cauchy Schwarz inequality, 31, 38Cayley Hamilton theorem, 139, 286, 331characteristic equation, 248characteristic polynomial, 139characteristic value, 248chemical reactions

balancing, 68classical adjoint, 115cofactor, 109, 110, 136cofactor matrix, 110

column rank, 155column space, 155companion matrix, 370complex conjugate, 5complex eigenvalues, 266complex numbers, 4complex numbers

arithmetic, 4roots, 7triangle inequality, 5

component, 40component of a force, 35components of a matrix, 82components of a vector, 18composition of linear transformations, 446condition number, 342conformable, 86conjugate

of a product, 13consistent, 67Coordinates, 15Cramer’s rule, 119, 139cross product, 41, 42

area of parallelogram, 42coordinate description, 43distributive law, 45, 47geometric description, 42

cross productcoordinate description, 43distributive law, 45geometric description, 42parallelepiped, 46

De Moivre’s theorem, 7defective, 255defective eigenvalue, 255derivative, 345determinant, 131

alternating property, 133cofactor, 107cofactor expansion, 136expanding along row or column, 108expansion along row (column), 136linear transformation, 444matrix inverse formula, 115, 137minor, 107nonzero, 177product, 112, 135product of eigenvalues, 329

478