Linear algebra a modern introduction, 4th edition

David Poole A M 'D E R N I N T R D U, C T I N 4th edition Fourth edition David Poole Trent University � � •- CENGAGE Learning· Australia• Brazil• Mexico• Singapore•United l0 * 0, (a) CA (c ) A > 0, * CA A > >0 > 0, + > 0, + 0, + 43 Hyperbola, x' = x, y' = y t , (x' )2/4 - (y' )2 /9 = + y, y ' -=j • �3 x ' -+ +-+-1 + +-+ + X -3 45 Parabola, x' = x - 2, y' = y 2, x' = -t(y ' )2 + y' y ::t::: ::t::: ::t::: ::t::: ::t::: y' � + + - x ' I + y + y 47.Ellipse,(x' )2/4 (y ' )2/ 12 = 49 Hyperbola, (x' )2 - (y' )2 = y 51 53 55 Answers to Selected Odd-Numbered Exercises Theorem ( c ) s h ows that i f vj = 0, t h en Qv Qv = Theorem ( b ) sh ows t h at ElHyperlipseb, ol(xa",)(2x/"50)2 - ((yy")" )22/=10 1= Degenerate (two lines) + y 13 57 Degenerate (a point) y �1 + + e +-_ x -2 59 Degenerate (two lines) y lS 17 61 63 65 67 2 Hyper b ol o i d of one s h eet , ( x ' ) ( y 3( ' ) ' ) z Hyper b ol i c par a bol o i d , = - (x ' ) (y ' ) = - V3(y ' ) V3(z ' ) Hyper b ol i c par a bol o i d , Ellipsoid, 3(x" )2 (y" )2 2(z")2 = = + + z + x' + (a) T [ - �j�11/6 l (c) 19 + • V; • { Qv1 , T (e) F (g) F Verify that (i) F 4], [ {[ -3 -2 QrQ = I • , Qvk } I]} -4 [ l [ �] } { [] uirm { [ -� l [ -lJ [ _ )] } (a) n -! �l Chapter 11 _ VectNot aorvectspaceor space; axiom fails VectVectNot aoorrvectssppaceaceor space; axiomVectComplfaoirlses.xpacevector space Exercises Review Q uestions • -2 -2 , doescons Hence,ists5.of unit vectors, becaus e i s an orthonormal set { [ �] } row(A): 0 rnl(A) { - � } [ [t] } null(A) : 01 , - 001 0 null(A') { = � ;· j Qvd { Qv1 , {v1 , , vk } 15 ANS21 ANS28 Answers to Selected Odd-Numbered Exercises 17.19 NotNot aa vectcomploresxpvectace; oaxir sopmsace;1,axi4, oandm faifailsl Nottaipvectlicatoironsparace;e nottheevenopertahtieosnsame.of addition and mul 25.29 NotSubsapacesubspace 27.31 NotSubsapacesubspace 33.Subs p ace 35 Subs p ace 37.41 NotSubsapacesubspace 43.39 NotSubsapacesubspace 45.47 TakeNot aUsutbso bepacethe x-axis and the y-axis, for example Then [ � ] and [ � ] are in U but [ � ] = [51 No� ] + [ � ] is not 53 Yesany; ss(calx)ar= + + ( + + for 55.57.NoYes; h(x) = + 59.No 61 Yes Linearly independent -l - l Linearly dependent; [ - ] = [ - 2 ] + [� � ] - [ _ � � ] 5.7 LiLinnearearllyy idependent ndependent; + = LiLinnearearllyy idependent ndependent; sin2 + cos2 13.17 LinearLinlyeardependent ; l n ( x ) = l n + l n ( x) l y i n dependent Li n ear l y dependent 19.23 NotBasisa basis 25 NotNot aa basbasiiss 29 [ -� ] 6 21 W U t (3 2t)p(x) j (x) g(x) W, t)q (x) tr(x) Exercises 6.2 3x = 11 x2 7x - 2(2x - x ) x (a) (b) 21 [p (x) ] u � · x · 35 dim V = 2, B = { l 37 dim V = B = { [ � � l [ � � l [ � � ] } 39.dim2 V=n)/2,B2 = { [ � � l [ � � ] } dim(Uthat V)if = dim U, +diisma basV is for then 43.41 (n -Show , i s a bas i s for� + + + 45 { l 47· { [ � � l [ � � l [ � - � l [ � � ] } 49.51 {{ ll , + 53.59 {sin2 cos=2 + = + + = 61.63 _(i) -_ 16x + _19 (i ) _ + - x, - x } 3, X (a) (b) {w , { ( w1 , w1 ) , x, • x • • • • • W, wJ (wn , wn )} x2 , l } x} - x, x - x } x, x} -x 4x - 3, ( a ) p0(x) i x - � x 3, p (x) i P (x) x - � x (c ) 3x x - 4x (pn l )(p n p)(p n p ) (p n p n - ) = [ � l = [ _:J, = [t _ tJ, = [� -�] U l [= : J H _ : n [ 1� �1 �1 i [ � l [ � l [- � �l = [� � ] Hl +J [: n H _: �- Exercises 6.3 [x] s Pc+- B [x] c [xJ , � [xJ , � Ps+-c Pc� u � Ps. c = [p(x) ] = - [p(x) ] c = - Pc+-s = Ps+-c [p(x)] � Pn �c � [p (x)] , Pc� B � Answers to Selected Odd-Numbered Exercises ANS29 and r (k[ :] ) = r [ �:J = = kT :] ([ ) Therefor-7e,]T is linear a ] ( ) T[ = T[ = (: ) (�) [-� l s = [ 2] l e = [ - 1/2] = 12 ( =2 ) Pc�s = [ � - � �� ],Ps�c = [ � - � J = = Let = Consider the effect of T and Don the standard [ - V3- 2\13)2 //22 ] [ - 3.1.232 ] bas i s for 2V3- ] [ 4646 ] [ 2V3 T) [ � ] = [� -! l T) [;] = [2; ] = { [ = � l [!] } T [;] does not make sense 21 - 1)1) - - 1)1)22 1)3 == 1)1) , (T =] T) [;J = s( r [;] ) = s([ ; / ) = [ ] = [ x] LiLinnearear ttrraansnsfoforrmmatatiioonn Linear transformation NotLineara litnrearansfotrarmnsatfoiromn ation ( T o s) [;J = r(s[;] ) = r ([!: : �] ) = NotWe havea linear transformation [ ] [x] == Therinversefores.e, T = and T = so and T are === and = = Onlyy ((ii i)) iissiinnker(range(T).T) Therefore, is linear Similarly, c Onl r ([ :] [ �] ) = r [ ] ker( T) = { [ � � ] } , range( T) = { [ � �] } AlkerOnll (ofyT)(tih=iem) is arineker(in raTnge() T) = = = = r( [ :J ) r( [ �] ) range( T) = (ka) + (ka + kb)x k (a + (a + b)x) 15 a , _ , [j (x) 1 [j (x) - 14x - 8x , 7b a x+ b (3 ( + (b ) (a) + ) = = 598 54 17 a = T(E 1 ) , b d T(E22 ) 23 Hint: 19 p ' (x + p ' (x + S) (p (x)) _ x 4Y 4(x - y) + ( - 3x + 4y) 3(x - y) + ( - 3x + 4y) 11 13 -y 2x + y · S) (p (x + l )) ' 29 (S Exercises 0 + (x + T(E2 ), (S o 27 (S T) (p (x)) 5(x - 3(x + T(E ) , c rzfn ( - 8(x + 3(x + x2 + 3x + 5x , T(a + bx + cx ) 3a - b - c x a + ex + 25 (S o 15 B b 17 T(4 - x + 3x ) 19 Hint: l3 a + 3b y (4x + y) - (3x + y) - 3(4x + y) + 4(3x + y) y S ( (p + q ) (x)) x ((p + q )(x)) x (p (x) + q (x)) xp (x) + x q (x) S (p (x)) + S ( q (x)) S (cp (x)) S((cp) (x)) x ( (cp) (x)) x(cp (x)) cxp (x) cS (p (x)) S a+ + b+d (a + c) + ((a + c) + (b + d))x (a + (a + b)x) + (c + (c + d)x) S (p (x) + q (x)) S I S I, S = = = + Exercises (ba ) ( ) (c ) ( a) (h) (c) {a + bx + cx : a - c, b {t + tx - tx }, IR - c} Answers to Selected Odd-Numbered Exercises ANS30 { [ � � l [ � �]} { [� �l [� �] } ; A basis for ker (T) is for range ( T) is , and a basis rank(T) = nullity(T) = 2, and rank( T) + nullity( T) = = dim M22 • A basis for ker ( T ) is { l + x - x }, and a basis for range ( T ) is { [ � ], [ � ] } ; rank( T) = 2, nullity(T) = and rank( T) + nullity( T) = = dim n, 43 transpose of, , - 60 1-, 559 unit lower triangular, 2-, 559 of a line, 36, unitarily diagonalizable, 546-547 7-, 559 o f a plane, , Parametric equation unitary, 545-546 compatible, 556 upper triangular, 162 Frobenius, 556 Partial pivoting, 84-85 zero, operator, 559 Partitioned matrix, 145- 49 Matrix-column representation of a matrix product, 146 Matrix factorization, 80 See also Singular value decomposition (SVD) Norm of a vector, 20, 535, 552 1-, 5 k-, 243 Path (s) 2-, 553 length of, 242 7-, 553 number of, 242-245 and diagonalization, 303-309 Euclidean, 553 L U, 80- 86 modified QR, 396-398 P1 L U, 86- QR, 392-394 Hamming, 554 and Schur's Triangularization Theorem, 408 Partial fractions, 1 simple, 242 Peano, Giuseppe, 429 max, 553 Penrose conditions, 586 sum, 552 Penrose, Roger, 603 taxicab, 530 Permutation matrix, uniform, 553 Perpendicular bisector, 33 Index Perron eigenvector, 33 Perron- Frobenius Theorem, 332-335 Perron, Oskar, 332 Perron root, 335 Perron's Theorem, 333 Petersen graph, 254 Pivot, 66 Pivoting, 66 partial, 84-85 Q QR algorithm, 398-399 QR factorization, 392-394 least squares and, 582-583 modified, 396-398 Quadratic equation (s), D6 graphing, 5-423 Quadratic form, 408-4 s Saddle point, 352 Scalar, Scalar matrix, Scalar multiple, 48 Scalar multiplication, - , 9, , 429 closure under, 92, 429 Scaling, indefinite, Schmidt, Erhard, 390 matrix associated with, 409 Schur complement, 283 Argand, Cl negative definite, Schur, Issai, complex, Cl negative semidefinite, Schur's Triangularization Theorem, 408 equation of, , 39, 41 positive definite, Schwarz, Karl Herman Amandus, positive semidefinite, Seidel, Philipp Ludwig, Plane, 38- 41 Polar decomposition, Polar form o f a complex number, C3-C6 Quadratic mean, 550 Seki Kowa, Takakazu, P6lya, George, A Quadric surface, 20 Set(s), A l -A4 Polynomial, D l - D l O Quotient of complex numbers, characteristic, 292 constant, D degree of, D irreducible, D7 Lagrange, 458 Legendre, 538 C2, C5 disjoint, A4 elements of, A empty, A2 R IR, IR3, IR", 9- 1 intersection of, A4 subset of, A2 union of, A4 Shifted inverse power method, 8- Taylor, 472 Racetrack game, - Similar matrices, -303, 508 trigonometric, Range, 2 , 482 Simple path, 242 zero of, D4 Rank Singular value decomposition (SVD ) , 590-599 Population distribution vector, 39 of a linear transformation, 484 applications of, 599-606 Population growth, 239-24 , 330-332 of a matrix, 72, 204 and condition number, 602 Positive definite matrix, singular value decomposition, 600 and least squares approximation, 603-605 quadratic form of, Rank Theorem, 72, 205, 386, 486 and matrix norms, 600-602 Positive matrix, 325 Ranking vector, 356-358 outer product form of, 596 Positive semidefinite matrix, Rational Roots Theorem, D5 and polar decomposition, Rayleigh, B aron, and pseudoinverse, 602-603 quadratic form of, Power method, 1 - Rayleigh quotient, and rank, 600 inverse, 7- Real axis, C l Singular values, 590- shifted, 6-3 Real part o f a complex Singular vectors, number, C l Size of a matrix, shifted inverse, - Predator-prey mo del, 343 Price vector (s), 235 15 Recurrence relation, 336 solution of, 337 Skew lines, Skew-symmetric matrix, 62 Primitive matrix, 335 Reduced r o w echelon for m, 73 Principal argument of a complex number, C4 Reducible matrix, 334 of a differential equation, Principal Axes Theorem, 1 Reflection, least squares, 4-582 Probability vector, Regular graph, of a linear system, 59 Product Regular matrix, minimum length least squares, 60 Solution of complex numbers, C l -C2 Repeller, of a recurrence relation, 337 of matrices, - 143 Resolving a vector, of a system of differential equations, 340-342 of polynomials, D2-D3 Resultants, 50 Span, 90, 56, 93, 438 Right singular vectors, 593 Spanning set of vectors, 88-92 Robotics, 226-229 Spanning sets, 438 -44 Production vector, 236 Projection orthogonal, 382-387, 538 Root, for a polynomial equation, D4 Spectral decomposition, 405 into a subspace, 382 Root mean square error, Spectral Theorem, 403 onto a vector, 27-28 Rotation, 6-2 Projection form of the Spectral Theorem, 405 center of, projection form of, 405 Spectrum, 403 Projection matrix, -2 9, 366, 586 Rotational symmetry, Spiral attractor, 355 Proof Roundoff error, , 83-84 Spiral repeller, 355 by contradiction, AS Row echelon form, 65 Square matrix, 9, 374 by contrapositive, AS Row equivalent matrices, 68 Square root of a matrix, 424 direct, A7 Row matrix, Standard basis, 198, 447 indirect, A7 Row-matrix representation of a matrix Standard matrix, by mathematical induction, B l -B Pseudoinverse of a matrix, 585-586, 602-603 Pythagoras' Theorem, 26, 537 product, 146 Row reduction, 66 Standard position, Standard unit vectors, 22 Row space, State vector, Row vector, 3, Steady state vector, 3 Index 16 span of, 438 Stochastic matrix, 232 Trigonometric polynomial, Strictly diagonally dominant matrix, 324 Triple scalar product identity, spanning sets of, 88-92 Strutt, John William, Trivial subspace, 437 state, Submatrices, 145 Turing, Alan Mathison, steady-state, 3 ternary, Subset, A2 u Subspace(s), 92, 433-438 unit, , 5 zero, 4, 429 Uniform norm, 553 fundamental, 380 spanned by a set of vectors, 92 - 93, 44 Union of sets, A4 sum of, 442 Unit circle, trivial, 43 Unit lower triangular matrix, zero, 437 Unit sphere, 53 Unit vector, 1, 535 Subtraction Vector form of the equation of a line, 36, Vector form of the equation o f a plane, 39, Vector space(s), 429 of complex numbers, C2 Unitarily diagonalizable matrix, 546-547 basis for, 446 of matrices, 140 Unitary matrix, 545-546 complex, 429, 432, 543-544 of polynomials, D2 Upper triangular matrix, 162 dimension of, 453 finite- dimensional, 453 block, 283 of vectors, 8, 433 infinite- dimensional, 453 Sum of complex numbers, C l of linear transformations, 48 v isomorphic, 493-495 subspace of, 433-438 Vandermonde, Alexandre-Theophile, of matrices, 140 Vandermonde determinant, of polynomials, D2 Vector(s), 3, 9, 429, 439 over "ll , 429, 432 P Venn diagram, A2-A3 of subspaces, 442 addition of, 5-6, 9, 439 Venn, John, A2 of vectors, 5-6, 9, 439 algebraic properties of, 10 Vertex of a graph, 242 Sum norm, 552 angle between, 24-26 Summation notation, A4-A7 binary, 14 Sustainable harvesting policy, 360 column, 3, Sylvester, James Joseph, 206, 280 complex, 429, , 543- 544 Weighted dot product, 532 Symmetric matrix, - 52, 60- complex dot product of, 543 Well- conditioned matrix, System of linear differential equations, 340-348 components of, Wessel, Caspar, C l System(s) of linear equations coordinate, 208, 448-452 Wey!, Hermann, 429 cross product of, 48-49, 286-287 Wheatstone bridge circuit, - 06 system(s) See Linear w Weight of a magic square, 460 demand, 36 Wilson, Edwin B , 49 T direction, 35 , 39 Wronskian, 457 Tail of a vector, distance between, 23-24 , 535 Taussky-Todd, Olga, 320 dot product of, -20 Taxicab circle, 530 equality of, x- axis, x Taxicab distance, 529-5 force, 50-53 xy-plane, Taxicab norm, 530 inner product of, xz-plane, Taxicab perpendicular bisector, 530 length of, , 5 Taxicab pi, 530 linear combination of, , 433 Taylor polynomial, 472 linearly dependent, 92-93, 443, 446 y- axis, Terminal point of a vector, linearly independent, 92-97, 443, 446 yz-plane, Ternary vector, norm of, 20, 5 , 552 y z Theorem, normal, 34, Tiling, 5 orthogonal, 26, 369- 373, 535, 537 Tournament, 244 orthonormal, 372, 537 "ll , , Trace of a matrix, parallel, "ll � , Transformation, 2 population distribution, 240 "ll m > "ll , linear, 3-2 4, 472-474 price, "ll �, matrix, 1 -2 6, 472 probability, Z-axis, Transitional matrix, production, 236 Zero matrix, Transitional probabilities, ranking, 356-358 Zero o f a polynomial, D Translational symmetry, resultant, 50 Zero subspace, 437 Transpose o f a matrix, , - 60 row, 3, Zero transformation, 474 Triangle inequality, 22, 540, 552 scalar multiplication of, - , 9, 429 Zero vector, 4, 429 ... Mathematical Association of America and the Canadian Mathematical Society I have also learned much from participation in the Canadian Mathematics Education Study Group and the Canadian Mathematics... two aspects of linear algebra that are scarcely ever mentioned to­ gether: finite linear algebra and numerical linear algebra By introducing modular arithmetic early, I have been able to make... Appendix C contains all the background material that is needed.) The cAs icon indicates that a computer algebra system (such as Maple, Mathematica, or MATLAB) or a calculator with matrix capa­