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Copyrighted Material JWST600-IND JWST600-Penney September 28, 2015 7:40 Printer Name: Trim: 6.125in × 9.25in INDEX Additive property of determinants, 244 Cofactor expansion, 240 Argument of a complex number, 297 Column space, 76 Column vector, 2 Band matrix, 407 Complementary vector, 319 Band width, 407 Complex eigenvalue, 302 Basis Complex eigenvector, 302 chain, 438 Complex linear transformation, 304 coordinate matrix for a basis, 217 Complex matrices, 301 coordinate transformation, 223 Complex number definition, 104 argument, 297 normalization, 315 conjugate, 302 ordered, 216 Complex vector space, 303 orthonormal, 315 Conjugate of a complex number, 302 point transformation, 223 Consumption matrix, 198 standard Coordinate matrix, 217 M(m,n), 119 Coordinates n, 119 coordinate vector, 216 Rn, 118 coordinate matrix for a basis, 217 COPYRIGHTEDcoordinate MATERIAL vector, 216 Cn, 302 point matrix, 217 Chain basis, 438 point transformation, 223 Characteristic polynomial, 275 Cramer’s rule, 265 Coefficient matrix, 75 Current, 41 Linear Algebra: Ideas and Applications, Fourth Edition. Richard C. Penney. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/penney/linearalgebra 487 JWST600-IND JWST600-Penney September 28, 2015 7:40 Printer Name: Trim: 6.125in × 9.25in 488 INDEX Data compression, 349 Haar function, 346 Determinant Hermitian additive property, 244 adjoint, 412 cofactor expansion, 240 symmetric, 414 Cramer’s rule, 265 Hermitian, orthogonal, 413 inverse matrix, 267 Homogeneous system, 80 Laplace expansion, 240 Householder matrix, 362 row exchange property, 241 Ij, 117 scalar property, 242 Ill-conditioned systems, 62, 448 uniqueness, 255 Image of a transformation, 152 Diagonal Inconsistent system, 35 kth, 176 Invariant subspace, 389 main, 176 Inverse Diagonal matrix, 176 formula for matrix inverse, 267 Diagonalizable over R, 278 Inverse iteration, 456 Diagonalization of a matrix, Invertible matrix, 187 288 Invertible transformation, 182 Discrete Haar Transform, 368 Dot product, 310 Kirchhoff’s laws, current law, 43 Kirchhoff’s laws, voltage law, 43 Echelon form, 48 Eigenbasis, 278 Least-squares solution, 371 Eigenspace Legrange Interpolation, 343 definition, 273 Line in Rn,38 Eigenvalue Linearly dependent complex, 302 dependency definition, 273 equation, 99 multiplicity, 278 system, 102 Eigenvector Linearly independent complex, 302 dependency definition, 273 equation, 99 Element of a set, 8 system, 102 Elementary matrix, 160 Equilibrium vector, 285 M(mn), 1 Equivalent linear systems, 29 Mc(m, n), 301 Matrix (R), 13 band, 407 Fast Haar Transform, 353 band width, 407 Fast Wavelet Transform, 353 characteristic polynomial, 275 Field, 301 column space, 76 Field properties, 301 condition number, 450 Flop, 68 diagonal, 176, 288 diagonalizable, 278 Gaussian elimination, 30 echelon form, 48 Gram-Schmidt elementary, 160 basis, 323 formula for inverse, 267 process, 323 Householder, 362 JWST600-IND JWST600-Penney September 28, 2015 7:40 Printer Name: Trim: 6.125in × 9.25in INDEX 489 identity, 118 Partitioned matrix, 172 inverse, 184 Pivoting invertible, 187 full, 62 LU decomposition, 206 partial, 62 nonsingular, 140 Plane in Rn,38 nullity, 139 Point matrix, 217 nullspace, 80 ProjP(Y), 319 of a transformation, 220 Projection orthogonal, 356 of a vector onto a vector, 319 partially orthogonal, 397 orthogonal compliment, 320 partitioned, 172 product, 166 QR decomposition, 326 product of a matrix with a vector, Quadratic form 78 defining matrix, 385 projection, 376 quadratic variety, 385 reduced form, 54 standard form, 381, 385 row reduced echelon form, 54 sparse, 189 Rank symmetric, 384 of a matrix, 136 transformation, 151 of a system of equations, 35 type-2, 205 Rayleigh quotient, 455 unipotent, 206 Reduced echelon form, 54 unitary, 412 Reduced singular value decomposition, upper Hessenberg, 409 397 upper triangular, 92 resistor, 42 Multiplication Row exchange property of determinants, matrix times a vector, 78 241 Multiplicity of an eigenvalue, 278 Row space, 133 Row vector, 2 Norm Euclidean norm, 447 S⟂, 320 infinity-norm, 447 Scalar product, functions, 335 one-norm, 447 Scalar product, space, 339 Normalization, 315 Scalar property of determinants, 242 null A, 139 Singular values, 399 Nullspace Spaces matrix, 80 ((−1, 1]), 335 (R), 13 OrthP(Y), 319 M(mn), 1 Orthogonal n,93 Hermitian, 413 Sparse matrix, 189 matrix, 356 Standard basis set, 312 M(m, n) vectors, 311 ordered, 225 Orthonormal set, 315 n, 119 M(m,n), 119 n,93 Subspace Partially orthogonal matrix, 397 subspace properties, 83 JWST600-IND JWST600-Penney September 28, 2015 7:40 Printer Name: Trim: 6.125in × 9.25in 490 INDEX Symmetric matrix, 384 matrix, 151 System matrix of a transformation, 220 differential equations, 294 target space, 151 homogeneous, 80 Transformations ill-conditioned, 62 composite, 165 least-squares solution, 371 Type-2 matrix, 205 rank, 35 Unitary matrix, 412 Target space, 151 Transformation Vector space domain, 151 complex, 303 elementary, 160 definition, 12 identity, 154 properties, 12 image, 152 Voltage, 41 invertible, 182 Voltage drop, 42.
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