## linear algebra 4/e

No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. (a) t2 + 1 and (c) t2 − t + 1 β= 588 1 0 0 −1 ⎜ 1 0 t2 C = ⎜ ⎝0 0 0 0 ⎛ 0 −1 0 ⎜1 1 0 ⎜ C=⎝ 0 0 0 0 0 1 0 0 0 0 , , 0 −1 0 0 ⎞ 0 0⎟ ⎟; β = {1, x, −2x + x2 , −3x + x3 } 0⎠ 0 ⎞ 0 0⎟ ⎟ −1⎠ 1 1 0 0 , 0 0 −1 0 0 0 0 Index 589 Index Absolute value of a complex number, 558 Absorbing Markov chain, 304 Absorbing state, 304 Addition of matrices, 9 Addition of vectors, 6 Additive function, 78 Additive inverse of an element of a ﬁeld, 553 of a vector, 12 Adjoint of a linear operator, 358–360 of a linear transformation, 367 of a matrix, 331, 359–360 uniqueness, 358 Algebraic multiplicity of an eigenvalue, see Multiplicity of an eigenvalue Algebraically closed ﬁeld, 482, 561 Alternating n-linear function, 239 Angle between two vectors, 202, 335 Annihilator of a subset, 126 of a vector, 524, 528 Approximation property of an orthogonal projection, 399 Area of a parallelogram, 204 Associated quadratic form, 389 Augmented matrix, 161, 174 Auxiliary polynomial, 131, 134, 137– 140 Axioms of the special theory of relativity, 453 Axis of rotation, 473 Back substitution, 186 Backward pass, 186 Basis, 43–49, 60–61, 192–194 cyclic, 526 dual, 120 Jordan canonical, 483 ordered, 79 orthonormal, 341, 346–347, 372 rational canonical, 526 standard basis for Fn , 43 standard basis for Pn (F ), 43 standard ordered basis for Fn , 79 standard ordered basis for Pn (F ), 79 uniqueness of size, 46 Bessel’s inequality, 355 Bilinear form, 422–433 diagonalizable, 428 diagonalization, 428–435 index, 444 invariants, 444 matrix representation, 424–428 product with a scalar, 423 rank, 443 signature, 444 sum, 423 symmetric, 428–430, 433–435 vector space, 424 Cancellation law for vector addition, 11 Cancellation laws for a ﬁeld, 554 Canonical form Jordan, 483–516 rational, 526–548 for a symmetric matrix, 446 Cauchy–Schwarz inequality, 333 Cayley–Hamilton theorem for a linear operator, 317 for a matrix, 318, 377 Chain of sets, 59 Change of coordinate matrix, 112– 115 Characteristic of a ﬁeld, 23, 41, 42, 430, 449, 555 Characteristic polynomial, 373 590 Index of a linear operator, 249 of a matrix, 248 Characteristic value, see Eigenvalue Characteristic vector, see Eigenvector Classical adjoint of an n × n matrix, 231 of a 2 × 2 matrix, 208 Clique, 94, 98 Closed model of a simple economy, 176–178 Closure under addition, 17 under scalar multiplication, 17 Codomain, 551 Coeﬃcient matrix of a system of linear equations, 169 Coeﬃcients Fourier, 119 of a diﬀerential equation, 128 of a linear combination, 24, 43 of a polynomial, 9 Cofactor, 210, 232 Cofactor expansion, 210, 215, 232 Column of a matrix, 8 Column operation, 148 Column sum of matrices, 295 Column vector, 8 Companion matrix, 526 Complex number, 556–561 absolute value, 558 conjugate, 557 fundamental theorem of algebra, 482, 560 imaginary part, 556 real part, 556 Composition of functions, 552 of linear transformations, 86– 89 Condition number, 469 Conditioning of a system of linear equations, 464 Congruent matrices, 426, 445, 451 Conic sections, 388–392 Conjugate linear property, 333 Conjugate of a complex number, 557 Conjugate transpose of a matrix, 331, 359–360 Consistent system of linear equations, 169 Consumption matrix, 177 Convergence of matrices, 284–288 Coordinate function, 119–120 Coordinate system left-handed, 203 right-handed, 202 Coordinate vector, 80, 91, 110– 111 Corresponding homogeneous system of linear equations, 172 Coset, 23, 109 Cramer’s rule, 224 Critical point, 439 Cullen, Charles G., 470 Cycle of generalized eigenvectors, 488–491 end vector, 488 initial vector, 488 length, 488 Cyclic basis, 526 Cyclic subspace, 313–317 Degree of a polynomial, 10 Determinant, 199–243 area of a parallelogram, 204 characterization of, 242 cofactor expansion, 210, 215, 232 Cramer’s rule, 224 of an identity matrix, 212 of an invertible matrix, 223 of a linear operator, 258, 474, 476–477 of a matrix transpose, 224 of an n × n matrix, 210, 232 n-dimensional volume, 226 properties of, 234–236 of a square matrix, 367, 394 of a 2 × 2 matrix, 200 uniqueness of, 242 of an upper triangular matrix, 218 Index volume of a parallelepiped, 226 Wronskian, 232 Diagonal entries of a matrix, 8 Diagonal matrix, 18, 97 Diagonalizable bilinear form, 428 Diagonalizable linear operator, 245 Diagonalizable matrix, 246 Diagonalization of a bilinear form, 428–435 problem, 245 simultaneous, 282, 325, 327, 376, 405 of a symmetric matrix, 431–433 test, 269, 496 Diagonalize, 247 Diﬀerentiable function, 129 Diﬀerential equation, 128 auxiliary polynomial, 131, 134, 137–140 coeﬃcients, 128 homogeneous, 128, 137–140, 523 linear, 128 nonhomogeneous, 142 order, 129 solution, 129 solution space, 132, 137–140 system, 273, 516 Diﬀerential operator, 131 null space, 134–137 order, 131, 135 Dimension, 47–48, 50–51, 103, 119, 425 Dimension theorem, 70 Direct sum of matrices, 320–321, 496, 545 of subspaces, 22, 58, 98, 275– 279, 318, 355, 366, 394, 398, 401, 475–478, 494, 545 Disjoint sets, 550 Distance, 340 Division algorithm for polynomials, 562 Domain, 551 Dominance relation, 95–96, 99 Dot diagram of a Jordan canonical form, 498– 500 591 of a rational canonical form, 535– 539 Double dual, 120, 123 Dual basis, 120 Dual space, 119–123 Economics, see Leontief, Wassily Eigenspace generalized, 485–491 of a linear operator or matrix, 264 Eigenvalue of a generalized eigenvector, 484 of a linear operator or matrix, 246, 371–374, 467–470 multiplicity, 263 Eigenvector generalized, 484–491 of a linear operator or matrix, 246, 371–374 Einstein, Albert, see Special theory of relativity Element, 549 Elementary column operation, 148, 153 Elementary divisor of a linear operator, 539 of a matrix, 541 Elementary matrix, 149–150, 159 Elementary operation, 148 Elementary row operation, 148, 153, 217 Ellipse, see Conic sections Empty set, 549 End vector of a cycle of generalized eigenvectors, 488 Entry of a matrix, 8 Equality of functions, 9, 551 of matrices, 9 of n-tuples, 8 of polynomials, 10 of sets, 549 Equilibrium condition for a simple economy, 177 Equivalence relation, 107, 551 congruence, 449, 451 592 Index unitary equivalence, 394, 472 Equivalent systems of linear equations, 182–183 Euclidean norm of a matrix, 467– 470 Euler’s formula, 132 Even function, 15, 21, 355 Exponential function, 133–140 Exponential of a matrix, 312, 515 Extremum, see Local extremum Field, 553–555 algebraically closed, 482, 561 cancellation laws, 554 characteristic, 23, 41, 42, 430, 449, 555 of complex numbers, 556–561 product of elements, 553 of real numbers, 549 sum of elements, 553 Field of scalars, 6–7, 47 Finite-dimensional vector space, 46– 51 Fixed probability vector, 301 Forward pass, 186 Fourier, Jean Baptiste, 348 Fourier coeﬃcients, 119, 348, 400 Frobenius inner product, 332 Function, 551–552 additive, 78 alternating n-linear, 239 codomain of, 551 composite, 552 coordinate function, 119–120 diﬀerentiable, 129 domain of, 551 equality of, 9, 551 even, 15, 21, 355 exponential, 133–140 image of, 551 imaginary part of, 129 inverse, 552 invertible, 552 linear, see Linear transformation n-linear, 238–242 norm, 339 odd, 21, 355 one-to-one, 551 onto, 551 polynomial, 10, 51–53, 569 preimage of, 551 range of, 551 real part of, 129 restriction of, 552 sum of, 9 vector space, 9 Fundamental theorem of algebra, 482, 560 Gaussian elimination, 186–187 back substitution, 186 backward pass, 186 forward pass, 186 General solution of a system of linear equations, 189 Generalized eigenspace, 485–491 Generalized eigenvector, 484–491 Generates, 30 Generator of a cyclic subspace, 313 Geometry, 385, 392, 436, 472–478 Gerschgorin’s disk theorem, 296 Gram–Schmidt process, 344, 396 Gramian matrix, 376 Hardy–Weinberg law, 307 Hermitian operator or matrix, see Self-adjoint linear operator or matrix Hessian matrix, 440 Homogeneous linear diﬀerential equation, 128, 137–140, 523 Homogeneous polynomial of degree two, 433 Homogeneous system of linear equations, 171 Hooke’s law, 128, 368 Householder operator, 397 Identity element in C, 557 in a ﬁeld, 553, 554 Identity matrix, 89, 93, 212 Identity transformation, 67 Ill-conditioned system, 464 Index Image, see Range Image of an element, 551 Imaginary number, 556 Imaginary part of a complex number, 556 of a function, 129 Incidence matrix, 94–96, 98 Inconsistent system of linear equations, 169 Index of a bilinear form, 444 of a matrix, 445 Inﬁnite-dimensional vector space, 47 Initial probability vector, 292 Initial vector of a cycle of generalized eigenvectors, 488 Inner product, 329–336 Frobenius, 332 on H, 335 standard, 330 Inner product space complex, 332 H, 332, 343, 348–349, 380, 399 real, 332 Input–output matrix, 177 Intersection of sets, 550 Invariant subspace, 77–78, 313– 315 Invariants of a bilinear form, 444 of a matrix, 445 Inverse of a function, 552 of a linear transformation, 99– 102, 164–165 of a matrix, 100–102, 107, 161– 164 Invertible function, 552 Invertible linear transformation, 99– 102 Invertible matrix, 100–102, 111, 223, 469 Irreducible polynomial, 525, 567– 569 Isometry, 379 Isomorphic vector spaces, 102–105 593 Isomorphism, 102–105, 123, 425 Jordan block, 483 Jordan canonical basis, 483 Jordan canonical form dot diagram, 498–500 of a linear operator, 483–516 of a matrix, 491 uniqueness, 500 Kernel, see Null space Kronecker delta, 89, 335 Lagrange interpolation formula, 51– 53, 125, 402 Lagrange polynomials, 51, 109, 125 Least squares approximation, 360– 364 Least squares line, 361 Left shift operator, 76 Left-handed coordinate system, 203 Left-multiplication transformation, 92–94 Legendre polynomials, 346 Length of a cycle of generalized eigenvectors, 488 Length of a vector, see Norm Leontief closed model, 176–178 open model, 178–179 Leontief, Wassily, 176 Light second, 452 Limit of a sequence of matrices, 284–288 Linear combination, 24–26, 28–30, 39 uniqueness of coeﬃcients, 43 Linear dependence, 36–40 Linear diﬀerential equation, 128 Linear equations, see System of linear equations Linear functional, 119 Linear independence, 37–40, 59– 61, 342 Linear operator, (see also Linear transformation), 112 adjoint, 358–360 characteristic polynomial, 249 594 Index determinant, 258, 474, 476–477 diagonalizable, 245 diagonalize, 247 diﬀerential, 131 diﬀerentiation, 131, 134–137 eigenspace, 264, 401 eigenvalue, 246, 371–374 eigenvector, 246, 371–374 elementary divisor, 539 Householder operator, 397 invariant subspace, 77–78, 313– 315 isometry, 379 Jordan canonical form, 483–516 left shift, 76 Lorentz transformation, 454–461 minimal polynomial, 516–521 nilpotent, 512 normal, 370, 401–403 orthogonal, 379–385, 472–478 partial isometry, 394, 405 positive deﬁnite, 377–378 positive semideﬁnite, 377–378 projection, 398–403 projection on a subspace, 86, 117 projection on the x-axis, 66 quotient space, 325–326 rational canonical form, 526–548 reﬂection, 66, 113, 117, 387, 472– 478 right shift, 76 rotation, 66, 382, 387, 472–478 self-adjoint, 373, 401–403 simultaneous diagonalization, 282, 405 spectral decomposition, 402 spectrum, 402 unitary, 379–385, 403 Linear space, see Vector space Linear transformation, (see also Linear operator), 65 adjoint, 367 composition, 86–89 identity, 67 image, see Range inverse, 99–102, 164–165 invertible, 99–102 isomorphism, 102–105, 123, 425 kernel, see Null space left-multiplication, 92–94 linear functional, 119 matrix representation, 80, 88– 92, 347, 359 null space, 67–69, 134–137 nullity, 69–71 one-to-one, 71 onto, 71 product with a scalar, 82 pseudoinverse, 413 range, 67–69 rank, 69–71, 159 restriction, 77–78 singular value, 407 singular value theorem, 406 sum, 82 transpose, 121, 126, 127 vector space of, 82, 103 zero, 67 Local extremum, 439, 450 Local maximum, 439, 450 Local minimum, 439, 450 Lorentz transformation, 454–461 Lower triangular matrix, 229 Markov chain, 291, 304 Markov process, 291 Matrix, 8 addition, 9 adjoint, 331, 359–360 augmented, 161, 174 change of coordinate, 112–115 characteristic polynomial, 248 classical adjoint, 208, 231 coeﬃcient, 169 cofactor, 210, 232 column of, 8 column sum, 295 companion, 526 condition number, 469 congruent, 426, 445, 451 conjugate transpose, 331, 359– 360 consumption, 177 Index 595 convergence, 284–288 determinant of, 200, 210, 232, 367, 394 diagonal, 18, 97 diagonal entries of, 8 diagonalizable, 246 diagonalize, 247 direct sum, 320–321, 496, 545 eigenspace, 264 eigenvalue, 246, 467–470 eigenvector, 246 elementary, 149–150, 159 elementary divisor, 541 elementary operations, 148 entry, 8 equality of, 9 Euclidean norm, 467–470 exponential of, 312, 515 Gramian, 376 Hessian, 440 identity, 89 incidence, 94–96, 98 index, 445 input–output, 177 invariants, 445 inverse, 100–102, 107, 161–164 invertible, 100–102, 111, 223, 469 Jordan block, 483 Jordan canonical form, 491 limit of, 284–288 lower triangular, 229 minimal polynomial, 517–521 multiplication with a scalar, 9 nilpotent, 229, 512 norm, 339, 467–470, 515 normal, 370 orthogonal, 229, 382–385 orthogonally equivalent, 384–385 permanent of a 2 × 2, 448 polar decomposition, 411–413 positive deﬁnite, 377 positive semideﬁnite, 377 product, 87–94 product with a scalar, 9 pseudoinverse, 414 rank, 152–159 rational canonical form, 541 reduced row echelon form, 185, 190–191 regular, 294 representation of a bilinear form, 424–428 representation of a linear transformation, 80, 88–92, 347, 359 row of, 8 row sum, 295 scalar, 258 self-adjoint, 373, 467 signature, 445 similarity, 115, 118, 259, 508 simultaneous diagonalization, 282 singular value, 410 singular value decomposition, 410 skew-symmetric, 23, 229, 371 square, 9 stochastic, see Transition matrix submatrix, 230 sum, 9 symmetric, 17, 373, 384, 389, 446 trace, 18, 20, 97, 118, 259, 281, 331, 393 transition, 288–291, 515 transpose, 17, 20, 67, 88, 127, 224, 259 transpose of a matrix inverse, 107 transpose of a product, 88 unitary, 229, 382–385 unitary equivalence, 384–385, 394, 472 upper triangular, 21, 218, 258, 370, 385, 397 Vandermonde, 230 vector space, 9, 331, 425 zero, 8 Maximal element of a family of sets, 58 Maximal linearly independent subset, 59–61 Maximal principle, 59 596 Index Member, see Element Michelson–Morley experiment, 451 Minimal polynomial of a linear operator, 516–521 of a matrix, 517–521 uniqueness, 516 Minimal solution to a system of linear equations, 364–365 Monic polynomial, 567–569 Multiplicative inverse of an element of a ﬁeld, 553 Multiplicity of an eigenvalue, 263 Multiplicity of an elementary divisor, 539, 541 n-dimensional volume, 226 n-linear function, 238–242 n-tuple, 7 equality, 8 scalar multiplication, 8 sum, 8 vector space, 8 Nilpotent linear operator, 512 Nilpotent matrix, 229, 512 Nonhomogeneous linear diﬀerential equation, 142 Nonhomogeneous system of linear equations, 171 Nonnegative vector, 177 Norm Euclidean, 467–470 of a function, 339 of a matrix, 339, 467–470, 515 of a vector, 333–336, 339 Normal equations, 368 Normal linear operator or matrix, 370, 401–403 Normalizing a vector, 335 Null space, 67–69, 134–137 Nullity, 69–71 Numerical methods conditioning, 464 QR factorization, 396–397 Odd function, 21, 355 One-to-one function, 551 One-to-one linear transformation, 71 Onto function, 551 Onto linear transformation, 71 Open model of a simple economy, 178–179 Order of a diﬀerential equation, 129 of a diﬀerential operator, 131, 135 Ordered basis, 79 Orientation of an ordered basis, 202 Orthogonal complement, 349, 352, 398–401 Orthogonal equivalence of matrices, 384–385 Orthogonal matrix, 229, 382–385 Orthogonal operator, 379–385, 472– 478 on R2 , 387–388 Orthogonal projection, 398–403 Orthogonal projection of a vector, 351 Orthogonal subset, 335, 342 Orthogonal vectors, 335 Orthonormal basis, 341, 346–347, 372 Orthonormal subset, 335 Parallel vectors, 3 Parallelogram area of, 204 law, 2, 337 Parseval’s identity, 355 Partial isometry, 394, 405 Pendular motion, 143 Penrose conditions, 421 Periodic motion of a spring, 127, 144 Permanent of a 2 × 2 matrix, 448 Perpendicular vectors, see Orthogonal vectors Physics Hooke’s law, 128, 368 pendular motion, 143 periodic motion of a spring, 144 special theory of relativity, 451– 461 Index spring constant, 368 Polar decomposition of a matrix, 411–413 Polar identities, 338 Polynomial, 9 annihilator of a vector, 524, 528 auxiliary, 131, 134, 137–140 characteristic, 373 coeﬃcients of, 9 degree of a, 10 division algorithm, 562 equality, 10 function, 10, 51–53, 569 fundamental theorem of algebra, 482, 560 homogeneous of degree two, 433 irreducible, 525, 567–569 Lagrange, 51, 109, 125 Legendre, 346 minimal, 516–521 monic, 567–569 product with a scalar, 10 quotient, 563 relatively prime, 564 remainder, 563 splits, 262, 370, 373 sum, 10 trigonometric, 399 unique factorization theorem, 568 vector space, 10 zero, 9 zero of a, 62, 134, 560, 564 Positive deﬁnite matrix, 377 Positive deﬁnite operator, 377–378 Positive semideﬁnite matrix, 377 Positive semideﬁnite operator, 377– 378 Positive vector, 177 Power set, 59 Preimage of an element, 551 Primary decomposition theorem, 545 Principal axis theorem, 390 Probability, see Markov chain Probability vector, 289 ﬁxed, 301 initial, 292 597 Product of a bilinear form and a scalar, 423 of complex numbers, 556 of elements of a ﬁeld, 553 of a linear transformation and scalar, 82 of matrices, 87–94 of a matrix and a scalar, 9 of a vector and a scalar, 7 Projection on a subspace, 76, 86, 98, 117, 398–403 on the x-axis, 66 orthogonal, 398–403 Proper subset, 549 Proper value, see Eigenvalue Proper vector, see Eigenvector Pseudoinverse of a linear transformation, 413 of a matrix, 414 Pythagorean theorem, 337 QR factorization, 396–397 Quadratic form, 389, 433–439 Quotient of polynomials, 563 Quotient space, 23, 58, 79, 109, 325–326 Range, 67–69, 551 Rank of a bilinear form, 443 of a linear transformation, 69– 71, 159 of a matrix, 152–159 Rational canonical basis, 526 Rational canonical form dot diagram, 535–539 elementary divisor, 539, 541 of a linear operator, 526–548 of a matrix, 541 uniqueness, 539 Rayleigh quotient, 467 Real part of a complex number, 556 of a function, 129 Reduced row echelon form of a matrix, 185, 190–191 598 Index Reﬂection, 66, 117, 472–478 of R2 , 113, 382–383, 387, 388 Regular transition matrix, 294 Relation on a set, 551 Relative change in a vector, 465 Relatively prime polynomials, 564 Remainder, 563 Replacement theorem, 45–46 Representation of a linear transformation by a matrix, 80 Resolution of the identity operator, 402 Restriction of a function, 552 of a linear operator on a subspace, 77–78 Right shift operator, 76 Right-handed coordinate system, 202 Rigid motion, 385–387 in the plane, 388 Rotation, 66, 382, 387, 472–478 Row of a matrix, 8 Row operation, 148 Row sum of matrices, 295 Row vector, 8 Rudin, Walter, 560 Saddle point, 440 Scalar, 7 Scalar matrix, 258 Scalar multiplication, 6 Schur’s theorem for a linear operator, 370 for a matrix, 385 Second derivative test, 439–443, 450 Self-adjoint linear operator or matrix, 373, 401–403, 467 Sequence, 11 Set, 549–551 chain, 59 disjoint, 550 element of a, 549 empty, 549 equality of, 549 equivalence relation, 107, 394, 449, 451 equivalence relation on a, 551 intersection, 550 linearly dependent, 36–40 linearly independent, 37–40 orthogonal, 335, 342 orthonormal, 335 power, 59 proper subset, 549 relation on a, 551 subset, 549 union, 549 Signature of a bilinear form, 444 of a matrix, 445 Similar matrices, 115, 118, 259, 508 Simpson’s rule, 126 Simultaneous diagonalization, 282, 325, 327, 376, 405 Singular value of a linear transformation, 407 of a matrix, 410 Singular value decomposition of a matrix, 410 Singular value decomposition theorem for matrices, 410 Singular value theorem for linear transformations, 406 Skew-symmetric matrix, 23, 229, 371 Solution of a diﬀerential equation, 129 minimal, 364–365 to a system of linear equations, 169 Solution set of a system of linear equations, 169, 182 Solution space of a homogeneous diﬀerential equation, 132, 137– 140 Space–time coordinates, 453 Span, 30, 34, 343 Special theory of relativity, 451– 461 axioms, 453 Lorentz transformation, 454–461 space–time coordinates, 453 Index time contraction, 459–461 Spectral decomposition, 402 Spectral theorem, 401 Spectrum, 402 Splits, 262, 370, 373 Spring, periodic motion of, 127, 144 Spring constant, 368 Square matrix, 9 Square root of a unitary operator, 393 Standard basis for Fn , 43 for Pn (F ), 43 Standard inner product on Fn , 330 Standard ordered basis for Fn , 79 for Pn (F ), 79 Standard representation of a vector space, 104–105 States absorbing, 304 of a transition matrix, 288 Stationary vector, see Fixed probability vector Statistics, see Least squares approximation Stochastic matrix, see Transition matrix Stochastic process, 291 Submatrix, 230 Subset, 549 linearly dependent, 36–40 linearly independent, 59–61 maximal linearly independent, 59–61 orthogonal, 335, 342 orthogonal complement of a, 349, 352, 398–401 orthonormal, 335 span of a, 30, 34, 343 sum, 22 Subspace, 16–19, 50–51 cyclic, 313–317 dimension of a, 50–51 direct sum, 22, 58, 98, 275–279, 318, 355, 366, 394, 398, 401, 599 475–478, 494, 545 generated by a set, 30 invariant, 77–78 sum, 275 zero, 16 Sum of bilinear forms, 423 of complex numbers, 556 of elements of a ﬁeld, 553 of functions, 9 of linear transformations, 82 of matrices, 9 of n-tuples, 8 of polynomials, 10 of subsets, 22 of vectors, 7 Sum of subspaces, (see also Direct sum, of subspaces), 275 Sylvester’s law of inertia for a bilinear form, 443 for a matrix, 445 Symmetric bilinear form, 428–430, 433–435 Symmetric matrix, 17, 373, 384, 389, 446 System of diﬀerential equations, 273, 516 System of linear equations, 25–30, 169 augmented matrix, 174 coeﬃcient matrix, 169 consistent, 169 corresponding homogeneous system, 172 equivalent, 182–183 Gaussian elimination, 186–187 general solution, 189 homogeneous, 171 ill-conditioned, 464 inconsistent, 169 minimal solution, 364–365 nonhomogeneous, 171 solution to, 169 well-conditioned, 464 T-annihilator, 524, 528 T-cyclic basis, 526 600 Index T-cyclic subspace, 313–317 T-invariant subspace, 77–78, 313– 315 Taylor’s theorem, 441 Test for diagonalizability, 496 Time contraction, 459–461 Trace of a matrix, 18, 20, 97, 118, 259, 281, 331, 393 Transition matrix, 288–291, 515 regular, 294 states, 288 Translation, 386 Transpose of an invertible matrix, 107 of a linear transformation, 121, 126, 127 of a matrix, 17, 20, 67, 88, 127, 224, 259 Trapezoidal rule, 126 Triangle inequality, 333 Trigonometric polynomial, 399 Trivial representation of zero vector, 36–38 Union of sets, 549 Unique factorization theorem for polynomials, 568 Uniqueness of adjoint, 358 of coeﬃcients of a linear combination, 43 of Jordan canonical form, 500 of minimal polynomial, 516 of rational canonical form, 539 of size of a basis, 46 Unit vector, 335 Unitary equivalence of matrices, 384–385, 394, 472 Unitary matrix, 229, 382–385 Unitary operator, 379–385, 403 Upper triangular matrix, 21, 218, 258, 370, 385, 397 Vandermonde matrix, 230 Vector, 7 additive inverse of a, 12 annihilator of a, 524, 528 column, 8 coordinate, 80, 91, 110–111 ﬁxed probability, 301 Fourier coeﬃcients, 119, 348, 400 initial probability, 292 linear combination, 24 nonnegative, 177 norm, 333–336, 339 normalizing, 335 orthogonal, 335 orthogonal projection of a, 351 parallel, 3 perpendicular, see Orthogonal vectors positive, 177 probability, 289 product with a scalar, 8 Rayleigh quotient, 467 row, 8 sum, 7 unit, 335 zero, 12, 36–38 Vector space, 6 addition, 6 basis, 43–49, 192–194 of bilinear forms, 424 of continuous functions, 18, 67, 119, 331, 345, 356 of cosets, 23 dimension, 47–48, 103, 119, 425 dual, 119–123 ﬁnite-dimensional, 46–51 of functions from a set into a ﬁeld, 9, 109, 127 inﬁnite-dimensional, 47 of inﬁnitely diﬀerentiable functions, 130–137, 247, 523 isomorphism, 102–105, 123, 425 of linear transformations, 82, 103 of matrices, 9, 103, 331, 425 of n-tuples, 8 of polynomials, 10, 86, 109 quotient, 23, 58, 79, 109 scalar multiplication, 6 of sequences, 11, 109, 356, 369 subspace, 16–19, 50–51 zero, 15 zero vector of a, 12 Index Volume of a parallelepiped, 226 Wade, William R., 439 Well-conditioned system, 464 Wilkinson, J. H., 397 Wronskian, 232 Z2 , 16, 42, 429, 553 Zero matrix, 8 601 Zero of a polynomial, 62, 134, 560, 564 Zero polynomial, 9 Zero subspace, 16 Zero transformation, 67 Zero vector, 12, 36–38 trivial representation, 36–38 Zero vector space, 15.

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