Linear Algebra

column rankfull, 127column space, 122combining rows, 4complement, A-7complementary subspaces, 132orthogonal, 260complex numbers, 355vector space over, 86component, 14composition, A-8self, 373computer algebra systems, 59–60concatenation, 130conditioning number, 67congruent figures, 288congruent plane figures, 288constant polynomial, 354contradiction, A-6contrapositive, A-3convex set, 180coordinateshomogeneous, 344with respect to a basis, 112corollary, A-1correspondence, 158, A-9coset, 190Cramer’s rule, 331–333cross product, 300crystals, 139–142diamond, 140graphite, 140salt, 139unit cell, 140da Vinci, Leonardo, 341degree of polynomial, 354determinant, 296, 301–319cofactor, 327Cramer’s rule, 332definition, 301exists, 310, 316Laplace expansion, 327minor, 327permutation expansion, 310, 313, 334diagonal matrix, 207, 224diagonalizable, 361–364difference equation, 419homogeneous, 419dilation, 161, 273representing, 199dimension, 117physical, 149dimensional constant, 149dimensional formula, 149direct map, 291direct sum, 128–136definition, 132external, 166internal, 166of two subspaces, 132direction vector, 35distance-preserving, 288division theorem, 354domain, A-8dot product, 40double precision, 67dual space, 190echelon form, 5free variable, 12leading variable, 5reduced, 47eigenspace, 368eigenvalue, eigenvectorof a matrix, 365of a transformation, 364element, A-6elementarymatrix, 225, 272elementary reduction operations, 4rescaling, 4row combination, 4swapping, 4elementary row operations, 4elimination, 2empty set, A-7entry, 13equivalenceclass, A-11canonical representative, A-11relation, A-10representative, A-11equivalence relation, A-10isomorphism, 167matrix equivalence, 242matrix similarity, 358row equivalence, 49equivalent statements, A-3Erlanger Program, 288