Linear Algebra

ight, 230, A-9two-sided, A-9inverse function, 230inverse image, 182inversion, 314isometry, 288isomorphism, 157–172characterized by dimension, 170definition, 158of a space with itself, 161Jordan block, 397Jordan form, 387–407represents similarity classes, 401kernel, 185Kirchhoff’s Laws, 70Klein, F., 288Laplace expansion, 326–330computes determinant, 327leading variable, 5least squares, 266–270lemma, A-1length, 39Leontief, W., 61linebest fit, 267in projective plane, 346line at infinity, 347line of best fit, 266–270lineartranspose operation, 127linear combination, 2Linear Combination Lemma, 52linear elimination, 2linear equation, 2coefficients, 2constant, 2homogeneous, 21inconsistent systems, 266satisfied by a vector, 15solution of, 2Gauss’s Method, 3Gauss-Jordan, 47solutions ofCramer’s rule, 332system of, 2linear mapdilation, 273reflection, 273, 291rotation, 272, 291seehomomorphism, 173skew, 274trace, 406linear mapsspace of, 177linear recurrence, 419linear recurrences, 418–425linear relationship, 98linear surface, 36linear transformationseetransformation, 176linearly dependent, 98linearly independent, 98LINPACK, 59magic square, 277normal, 278magic squares, 277–281map, A-8distance-preserving, 288extended linearly, 170restriction, 377self composition, 373Maple, 59Markov chain, 283historyless, 283Markov chains, 282–287Mathematica, 59mathematical induction, 23, A-5MATLAB, 59matrix, 13adjoint, 328antisymmetric, 135augmented, 14block, 244, 312change of basis, 236characteristic polynomial, 368cofactor, 327column, 13column space, 122conditioning number, 67determinant, 296, 301diagonal, 207, 224diagonalizable, 361diagonalized, 242elementary reduction, 225, 272entry, 13equivalent, 242