Linear Algebra

composition, A-9self, 361computer algebra systems, 62–63concatenation, 133conditioning number, 71congruent figures, 281congruent plane figures, 281contradiction, A-6contrapositive, A-3convex set, 177coordinateshomogeneous, 334with respect to a basis, 115corollary, A-1correspondence, 155, A-9coset, 187Cramer’s rule, 325–327cross product, 292crystals, 142–145diamond, 143graphite, 143salt, 142unit cell, 143da Vinci, Leonardo, 331determinant, 288, 293–312cofactor, 321Cramer’s rule, 326definition, 293exists, 303, 309Laplace expansion, 321minor, 321permutation expansion, 302, 306,328diagonal matrix, 204, 219diagonalizable, 350–353difference equation, 402homogeneous, 402dilation, 157, 270representing, 197dimension, 120physical, 146direct map, 284direct sum, 131), 139definition, 135external, 162internal, 162of two subspaces, 135direction vector, 35distance-preserving, 281division theorem, 344domain, A-8dot product, 40double precision, 69dual space, 188echelon form, 5free variable, 12leading variable, 5reduced, 47eigenspace, 356eigenvalue, eigenvectorof a matrix, 354of a transformation, 353element, A-7elementarymatrix, 221, 269elementary reduction operations, 4pivoting, 4rescaling, 4swapping, 4elementary row operations, 4empty set, A-8entry, 13equivalenceclass, A-12canonical representative, A-12relation, A-10representative, A-12equivalence relation, A-10, A-11isomorphism, 163matrix equivalence, 238matrix similarity, 347row equivalence, 50equivalent statements, A-3Erlanger Program, 281Euclid, 281even functions, 99, 137even polynomials, 394external direct sum, 162Fibonacci sequence, 401field, 140–141definition, 140finite-dimensional vector space, 119flat, 36form, 56free variable, 12full column rank, 130