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Instructor’s Solutions Manual

Elementary Linear Algebra with Applications Ninth Edition

Bernard Kolman Drexel University

David R. Hill Temple University

Editorial Director, Computer Science, Engineering, and Advanced Mathematics: Marcia J. Horton Senior Editor: Holly Stark Editorial Assistant: Jennifer Lonschein Senior Managing Editor/Production Editor: Scott Disanno Art Director: Juan L´ opez Cover Designer: Michael Fruhbeis Art Editor: Thomas Benfatti Manufacturing Buyer: Lisa McDowell Marketing Manager: Tim Galligan Cover Image: (c) William T. Williams, Artist, 1969 Trane, 1969 Acrylic on canvas, 108!! × 84!! . Collection of The Studio Museum in Harlem. Gift of Charles Cowles, New York.

" c 2008, 2004, 2000, 1996 by Pearson Education, Inc. Pearson Education, Inc. Upper Saddle River, New Jersey 07458 c 1991, 1986, 1982, by KTI; Earlier editions " 1977, 1970 by Bernard Kolman

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

Printed in the United States of America 10

9

8

7

6

5

4

3

2

1

ISBN 0-13-229655-1

Pearson Pearson Pearson Pearson Pearson Pearson Pearson Pearson

Education, Ltd., London Education Australia PTY. Limited, Sydney Education Singapore, Pte., Ltd Education North Asia Ltd, Hong Kong Education Canada, Ltd., Toronto Educaci´on de Mexico, S.A. de C.V. Education—Japan, Tokyo Education Malaysia, Pte. Ltd

Contents Preface

iii

1 Linear Equations and Matrices 1.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Algebraic Properties of Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Special Types of Matrices and Partitioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Matrix Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Computer Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 7 9 14 16 18 19 24

2 Solving Linear Systems 2.1 Echelon Form of a Matrix . . . . . 2.2 Solving Linear Systems . . . . . . . 2.3 Elementary Matrices; Finding A−1 2.4 Equivalent Matrices . . . . . . . . 2.5 LU -Factorization (Optional) . . . . Supplementary Exercises . . . . . . Chapter Review . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 30 32 33 33 35

3 Determinants 3.1 Def inition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cofactor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Other Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 37 39 41 42 42 43

4 Real Vector Spaces 4.1 Vectors in the Plane and in 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Span and Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Coordinates and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 47 48 51 52 54 56 58 62

. . . . . . .

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1

ii

CONTENTS Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Inner Product Spaces 5.1 Standard Inner Product on R2 and R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Cross Product in R3 (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Least Squares (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 69 71 71 74 77 81 84 85 86 90

6 Linear Transformations and Matrices 93 6.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Kernel and Range of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4 Vector Space of Matrices and Vector Space of Linear Transformations (Optional) . . . . . . . 99 6.5 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.6 Introduction to Homogeneous Coordinates (Optional) . . . . . . . . . . . . . . . . . . . . . . 103 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Eigenvalues and Eigenvectors 109 7.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Diagonalization and Similar Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.3 Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8 Applications of Eigenvalues and Eigenvectors (Optional) 129 8.1 Stable Age Distribution in a Population; Markov Processes . . . . . . . . . . . . . . . . . . . 129 8.2 Spectral Decomposition and Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 130 8.3 Dominant Eigenvalue and Principal Component Analysis . . . . . . . . . . . . . . . . . . . . 130 8.4 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.5 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.6 Real Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.7 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.8 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10 MATLAB Exercises

137

Appendix B Complex Numbers 163 B.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 B.2 Complex Numbers in Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Preface This manual is to accompany the Ninth Edition of Bernard Kolman and David R.Hill’s Elementary Linear Algebra with Applications. Answers to all even numbered exercises and detailed solutions to all theoretical exercises are included. It was prepared by Dennis Kletzing, Stetson University. It contains many of the solutions found in the Eighth Edition, as well as solutions to new exercises included in the Ninth Edition of the text.

Chapter 1

Linear Equations and Matrices Section 1.1, p. 8 2. x = 1, y = 2, z = −2. 4. No solution. 6. x = 13 + 10t, y = −8 − 8t, t any real number. 8. Inconsistent; no solution. 10. x = 2, y = −1. 12. No solution. 14. x = −1, y = 2, z = −2.

16. (a) For example: s = 0, t = 0 is one answer. (b) For example: s = 3, t = 4 is one answer. (c) s = 2t .

18. Yes. The trivial solution is always a solution to a homogeneous system. 20. x = 1, y = 1, z = 4. 22. r = −3. 24. If x1 = s1 , x2 = s2 , . . . , xn = sn satisfy each equation of (2) in the original order, then those same numbers satisfy each equation of (2) when the equations are listed with one of the original ones interchanged, and conversely. 25. If x1 = s1 , x2 = s2 , . . . , xn = sn is a solution to (2), then the pth and qth equations are satisfied. That is, ap1 s1 + · · · + apn sn = bp

aq1 s1 + · · · + aqn sn = bq . Thus, for any real number r, (ap1 + raq1 )s1 + · · · + (apn + raqn )sn = bp + rbq . Then if the qth equation in (2) is replaced by the preceding equation, the values x1 = s1 , x2 = s2 , . . . , xn = sn are a solution to the new linear system since they satisfy each of the equations.

2

Chapter 1 26. (a) A unique point. (b) There are infinitely many points. (c) No points simultaneously lie in all three planes. C2

28. No points of intersection:

One point of intersection:

Two points of intersection:

C2

C1

C1

C2

C1

C2

C1

C1 =C2

Infinitely many points of intersection:

30. 20 tons of low-sulfur fuel, 20 tons of high-sulfur fuel. 32. 3.2 ounces of food A, 4.2 ounces of food B, and 2 ounces of food C. 34. (a)

p(1) = a(1)2 + b(1) + c = a + b + c = −5 p(−1) = a(−1)2 + b(−1) + c = a − b + c = 1

p(2) = a(2)2 + b(2) + c = 4a + 2b + c = 7.

(b) a = 5, b = −3, c = −7.

Section 1.2, p. 19 

0 1  2. (a) A =  0  0 1

1 0 1 1 1

0 1 0 0 0

0 1 0 0 0

 1 1   0  0 0



0 1  (b) A =  1  1 1

1 0 1 0 0

1 1 0 1 0

1 0 1 0 0

 1 0   0 .  0 0

4. a = 3, b = 1, c = 8, d = −2.   ' ( 5 −5 8 7 −7   (b) Impossible. (c) . 6. (a) C + E = E + C = 4 2 9 . 0 1 5 3 4     −9 3 −9 0 10 −9 (e)  8 −1 −2 . (f) Impossible. (d)  −12 −3 −15 . −6 −3 −9 −5 −4 3 

1 8. (a) AT =  2 3

 ' 2 1 1 , (AT )T = 2 4

2 1

( 3 . 4



5 (b)  −5 8

4 2 9

 5 3 . 4

(c)

'

−6 11

( 10 . 17

3

Section 1.3

(d)

'

10. Yes: 2 



( 0 −4 . 4 0 '

1 0 0 1

(

3 (e)  6

4 . 3

'

(f)

'

(9 '10 ( 1 0 3 0 = . 0 0 0 2  −3 −3 . λ−4

+1

λ−1 −2 12.  −6 λ+2 −5 −2

2( . 6

17 −16

14. Because the edges can be traversed in either direction.   x1  x2    16. Let x =  .  be an n-vector. Then  ..  xn

       x1 0 x1 + 0 x1  x2   0   x2 + 0   x2          x + 0 =  .  +  .  =  .  =  .  = x.  ..   ..   ..   ..  xn 0 xn + 0 xn 

18.

n m ) ) i=1 j=1

aij = (a11 + a12 + · · · + a1m ) + (a21 + a22 + · · · + a2m ) + · · · + (an1 + an2 + · · · + anm ) = (a11 + a21 + · · · + an1 ) + (a12 + a22 + · · · + an2 ) + · · · + (a1m + a2m + · · · + anm ) n m ) ) aij . = j=1 i=1

19. (a) True.

n ) i=1

(b) True.

n ) i=1

(ai + 1) =

n )

ai +

i=1

n )

1=

i=1

  m n ) )  1 = m = mn. j=1

n )

ai + n.

i=1

i=1

  m m m m n ) ) ) ) ) bj  = a1 ai   (c) True.  bj bj + a2 bj + · · · + an 

i=1

j=1

j=1

j=1

j=1

m ) bj = (a1 + a2 + · · · + an ) j=1 . / n m n m ) ) ) ) ai bj bj = ai = i=1

20. “new salaries” = u + .08u = 1.08u.

Section 1.3, p. 30 2. (a) 4. 4. x = 5.

(b) 0.

(c) 1.

(d) 1.

j=1

j=1

i=1

4

Chapter 1 √ 6. x = ± 2, y = ±3. 8. x = ±5.

10. x = 65 , y = 12. (a)

14. (a) (d)

16. (a)

(f)

12 . 5

   8 8 15 −7 14 (c)  23 −5 Impossible. 29. (d)  14 13 . 13 9 13 −1 17 ( ' ( ' 28 8 38 58 12 . (b) Same as (a). (c) . 66 13 34 4 41 ( ' ( ' −16 −8 −26 28 32 ; same. (f) . Same as (c). (e) 16 18 −30 0 −31   −1 4 2 1 0 (d)  −2 (e) 10. 1. (b) −6. (c) −3 0 1 . 8 4 . 3 −12 −6   9 0 −3  0 (g) Impossible. 0 0 . −3 0 1 

 0 −1 1 (b)  12 5 17 . 19 0 22



(e) Impossible.

18. DI2 = I2 D = D. ( ' 0 0 . 20. 0 0     1 0  14  18    22. (a)  (b)   0 .  3 . 13 13             1 1 −2 −1 −2 −1 24. col1 (AB) = 1  2  + 3 4  + 2  3; col2 (AB) = −1  2 + 2  4  + 4  3 . 3

0

3

−2

0

−2

(b) BAT 1 0 1 28. Let A = aij be m × p and B = bij be p × n. 26. (a) −5.

0

(a) Let the ith row of A consist entirely of zeros, so that aik = 0 for k = 1, 2, . . . , p. Then the (i, j ) entry in AB is p ) aik bkj = 0 for j = 1, 2, . . . , n. k=1

(b) Let the jth column of A consist entirely of zeros, so that akj = 0 for k = 1, 2, . . . , m. Then the (i, j) entry in BA is m ) bik akj = 0 for i = 1, 2, . . . , m. k=1



2 3 30. (a)  2 0

3 −3 1 0 2 0 3 0 −4 0 1 1

 1 3 . 0 1



2 3 (b)  2 0

3 −3 1 0 2 0 3 0 −4 0 1 1

   x1   7 1   x 2     3 −2  .   x3  =  0    3   x4  5 1 x5

5

Section 1.3 2 3   (c) 2 0

3 −3 1 0 2 0 3 0 −4 0 1 1 ' (' ( ' ( −2 3 5 x1 32. = . 1 −5 x2 4 34. (a)

 1 7 3 −2   0 3  1 5

2x1 + x2 + 3x3 + 4x4 = 0 3x1 − x2 + 2x3 =3 −2x1 + x2 − 4x3 + 3x4 = 2

(b) same as (a). 

     3 1 −1 (b) x1  2 + x2  −1  =  −2 . 1 3 1      x1 0 1 2 1 (b)  1 1 2   x2 =  0 . 2 0 2 x3 0

( ' ( ' ( ' ' ( 2 1 4 3 . + x2 + x3 = 36. (a) x1 −2 1 −1 4

38. (a)

'

  ' ( ( x1 1 1 2 0   x2 = . 2 5 3 1 x3

39. We have

u·v =

n ) i=1

0 ui vi = u1 u2

 1 0 0 40. Possible answer:  2 0 0. 3 0 0 

42. (a) Can say nothing. 43. (a) Tr(cA) =

n ) i=1



 v1  1  v2  · · · un  .  = uT v.  ..  vn

(b) Can say nothing.

n ) aii = c Tr(A). caii = c i=1

n n n ) ) ) (b) Tr(A + B) = bii = Tr(A) + Tr(B). aii + (aii + bii ) = i=1

0 1 (c) Let AB = C = cij . Then

i=1

i=1

Tr(AB) = Tr(C) =

n )

cii =

n )

aTii =

i=1

n )

aijT aji =

j=1

Hence, Tr(AT A) ≥ 0.

n ) j=1

n ) n )

bki aik = Tr(BA).

k=1 i=1

aii = Tr(A).

i=1

0 1 (e) Let AT A = B = bij . Then bii =

n )

aik bki =

i=1 k=1

i=1

T (d) Since aii = aii , Tr(AT ) =

n n ) )

2 aji

=⇒

Tr(B) = Tr(AT A) =

n )

i=1

bii =

n n ) ) i=1 j=1

a2ij ≥ 0.

6

Chapter 1 44. (a) 4.

(b) 1.

(c) 3.

45. We have Tr(AB − BA) = Tr(AB) − Tr(BA) = 0, while Tr

(3 2' 1 0 = 2. 0 1 

 b1j  b2j  0 1 0 1  46. (a) Let A = aij and B = bij be m × n and n × p, respectively. Then bj =   .  and the ith  ..  bnj n ) aik bkj , which is exactly the (i, j) entry of AB . entry of Abj is k=1

(b) The ith row of AB is we have

04

k

aik bk1

ai b =

4

k

04

k

aik bk2 · · ·

aik bk1

This is the same as the ith row of Ab.

4

k

4

k

1 1 0 aik bkn . Since ai = ai1 ai2 · · · ain ,

aik bk2 · · ·

4

k

1 aik bkn .

0 1 0 1 47. Let A = aij and B = bij be m × n and n × p, respectively. Then the jth column of AB is  a11 b1j + · · · + a1n bnj   .. (AB )j =   . 

am1 b1j  a11  .. = b1j  .

am1

+ · · · + amn bnj  

 a1n   .   + · · · + bnj  ..  amn

= b1j Col1 (A) + · · · + bnj Coln (A).

Thus the jth column of AB is a linear combination of the columns of A with coefficients the entries in bj . 48. The value of the inventory of the four types of items. 50. (a) row 1 (A) · col1 (B) = 80(20) + 120(10) = 2800 grams of protein consumed daily by the males. (b) row 2 (A) · col2 (B) = 100(20) + 200(20) = 6000 grams of fat consumed daily by the females.

51. (a) No. If x = (x1 , x 2 , . . . , xn ), then x · x = x21 + x22 + · · · + xn2 ≥ 0. (b) x = 0. 52. Let a = (a1 , a 2 , . . . , a n ), b = (b1 , b2 , . . . , b n ), and c = (c1 , c2 , . . . , cn ). Then (a) a · b =

n )

ai bi and b · a =

n n n ) ) ) bi ci = a · c + b · c. (ai + bi )ci = ai ci + i=1

(c) (ka) · b =

bi ai , so a · b = b · a.

i=1

i=1

(b) (a + b) · c =

n )

n )

(kai )bi = k

i=1

i=1

n ) i=1

i=1

ai bi = k(a · b).