linear algebra inclass exam 12/04 7-9pm (california timezone)
nimab8
1_AnswerKey3rdEditionAPrtraitofLinearAlgebra.pdf
A Portrait of
Linear Algebra
Selected Answers
to the Exercises
Jude Thaddeus Socrates
Version Date: 3 January 2018
www.kendallhunt.com
Chapter Zero Exercises
1. A True logical statement. 2. A logical statement, but it is False, because −5 3 but 25 9. 3. A True logical statement, using the properties of inequalities found in Appendix A. 4. A False logical statement, because if x 0, then x is imaginary. 5. A True logical statement as of June 2009, with 237 consecutive weeks. 6. Not a logical statement, because it cannot be ascertained to be True or False (“best” is not
a well-defined adjective; unlike the previous Exercise, where “most number of consecutive weeks as number 1” is well defined).
7. Converse: If you can watch TV tonight, then you did your homework before dinner. Inverse: If you do not do your homework before dinner, you cannot watch TV tonight. Contrapositive: If you cannot watch TV tonight, then you did not do your homework before dinner.
8. Converse: If we don’t go to the beach tomorrow, then it rained. Inverse: If it doesn’t rain tomorrow, we will go to the beach. Contrapositive: If we go to the beach tomorrow, then it did not rain.
9. Converse: If cosx ≥ 0, then 0 ≤ x ≤ /2. Inverse: If x /2 or x 0, then cosx 0. Contrapositive: If cosx 0, then x /2 or x 0.
10. If fx is continuous on the closed interval a, b then it possesses both a maximum and a minimum on a, b. Converse: If fx possesses both a maximum and a minimum on a, b, then fx is continuous on a, b. Inverse: If fx is not continuous on a, b, then fx either does not possess an absolute maximum or an absolute minimum on a, b. Contrapositive: If fx does not possess either an absolute maximum or an absolute minimum on a, b, then fx is not continuous at x a.
11. A B a, b, c, f, g, h, i, j, m, p, q, A ∩ B c, h, j, A − B a, f, i, m, B − A b, g, p, q.
12. A B a, b, d, g, h, j, k, p, q, r, s, t, v, A ∩ B d, g, h, p, t, A − B a, j, r, B − A b, k, q, s, v.
23. If there were a largest positive number x, what can you say about x 1? 27. “If n does not have a prime factor which is at most n , then n is prime.” The number
11303 is composite. One prime factor is smaller than 100. 38. 2027 and 2029. 39. 233 49. Hint: In Step 3, write 2n1 as 22n 2n 2n. 54. f. For any two sets X and Y : X ∩ Y ⊆ X and X ∩ Y ⊆ Y. 55. a.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 58. a. ∅, a, b, c, a, b, a, c, b, c, a, b, c; 8 subsets.
c. you get exactly the same list as the subsets on the right column.
2 Selected Answers to the Exercises
Chapter One Exercises
1.1 Exercises
1. These are found in the Key Concepts. 2. b. ‖u‖ 65 ; c. u1 1
65 ⟨−4, 7 and u2 −1
65 ⟨−4, 7 d. 3v ⟨9, 15,
5w ⟨5,−10, v 5w ⟨14, 5 and 3v− 5w ⟨4, 25 3. b. 2u ⟨10,−6, 4, 3w ⟨−6, 15, 12, 2u 3w ⟨4, 9, 16 and 2u − 3w ⟨16,−21,−8
c. ‖w‖ 45 3 5 d. u1 1 3 5
⟨−2, 5, 4 and u1 −1 3 5
⟨−2, 5, 4.
e. i. − 3 5
w ⟨6/5,−3,−12/5 ii. 2u 5v ⟨30,−6,−31 iii. 3w − 4u ⟨−26, 27, 4 iv. −4u 7v− 2w ⟨12, 2,−65.
4. a. u v ⟨1,−2, 7, 3 b. u w ⟨−1,−3, 4,−2 c. v− w ⟨2, 1, 3, 5 d. − 2u ⟨−6, 10,−2,−14 e. 3
4 v − 3
2 , 9
4 , 9
2 ,−3 f. − 5
3 w 20
3 ,− 10
3 ,−5, 15
g. 5u 3v ⟨9,−16, 23, 23 h. − 3 2
u 5 4
v −7, 45 4
, 6,− 31 2
i. 2u − 3v 7w ⟨−16,−5, 5,−37 j. − 5u 2v− 4w ⟨−3, 23,−5,−7 k. − 3
2 u 3
4 v− 5
3 w 2
3 , 77
12 ,−2, 3
2 l. 3
2 u − 3
4 v 2w −2,− 23
4 , 3,− 9
2 5. u ⟨−15, 6, 7 and v ⟨42,−17,−16. 6. Yes: ⟨−3, 7 40⟨5,−2 29⟨−7, 3. 7. Yes: ⟨−17,−9, 29,−37 5⟨3,−5, 1, 7 8⟨−4, 2, 3,−9. 8. No: Using the first two coordinates, we get x −4 and y 9, but although these satisfy
the 3rd coordinate, they do not satisfy the 4th. 9. u ⟨−3, 4, 2, 6,−7 and v ⟨−1,−3, 5,−3, 2. 10. 7,−3 11. −4, 1, 7 12. u ⟨−4, 4,−8 22. Contrapositive: if u ⟨u1, u2 and v ⟨v1, v2 are vectors in 2, then they are not
parallel to each other if and only if u1v2 − u2v1 ≠ 0. 35. PQ is 26 cm. long.
1.2 Exercises
1. y 4x/7 2. y −5x/3 3. x 5t, y −4t, z 2t, and t x/5 y/−4 z/2. 4. x −t, y 3t, z −6t, and t −x y/3 z/−6. 5. 7x 5y 6 6. x 2 − 3t, y −7 6t, z 4 8t, and t x − 2
−3
y 7 6
z − 4 8
7. x 3 2t, y 2, z −5 − 5t. Not possible because the direction vector has 0 in the y-component.
8. v PQ ⟨4,−2, 3, so x −4 4t, y 3 − 2t, z −5 3t is one possible answer (other answers are possible).
9. 2x − 11y z 0. 10. 31x − 29y − 13z 0. 11. 10x − 2y 15z 0. We must solve for s from y, solve for r from z, then substitute these
into x. 12. Span⟨4,−10, 6,⟨−6, 15,−9 is only a line through the origin, because the vectors are
parallel to each other. 13. x y z 3 14. 9x 10y − 2z 28
Selected Answers to the Exercises 3
15. They determine a line because the vector AB is parallel to AC. 17. 3,−4, 7 satisfies the equation. If t 1, we get the point 7,−7, 13, which also satisfies
the equation. Since two points on the line are also on the plane, the whole line is on the plane. Alternatively, you can solve for x, y and z from the equation of the line, and substitute them into that of the plane, and get 0 0, showing that the equation of the plane is satisfied by every point on the line.
18. If Q 3, 4,−1, then PQ ⟨1,−1,−8 is not parallel to ⟨1,−2, 5, so P is not on L. Equation: 21x 13y z 114.
19. 7x 2y 4z 15 20. − 65 29
, 21 29
, 52 29
23. a. the point does not satisfy the symmetric equations; b. x − 5 3
y 2
5 −z 4
24. 13x − 7y 4z 95 25. 17x − 4y 22z −80 28. 28. a. 2x 6y 3z 0; b. w does not satisfy this equation. 33. d. 6x − 5y 4z 60; g. 3x − 2z 18 i. z −5 34. a. D at x0 − x12 bt y0 − y12 ct z0 − z12
b. dD dt
2t ax0 − x1 by0 − y1 cz0 − z1
c. t ax1 − x0 by1 − y0 cz1 − z0; d. d 2D
dt2 2 0.
e. the critical point is a local minimum by the 2nd derivative test; since D goes to positive infinity in both directions, the critical point is also an absolute maximum.
35. The critical value is t −3 66
; 53 11
,− 67 22
, 51 22
; distance: 7 22
374
36. The critical value is t 14 30
; 98 15
, 7 3
,− 46 15
; distance: 1 15
25530
1.3 Exercises
1. ‖u‖ 119 . 2. 2. cos 5/ 3161 and ≈ 1. 481 radians. 3. ‖2u 5v‖ 941 ≈ 32. 68, and ‖2u‖ ‖5v‖ 136 1625 ≈ 51. 97. The second
quantity should be bigger by the Triangle Inequality. 4. cos 37/ 6391 , so ≈ 1. 09 radians. 5. cos −15/ 7 23 , so ≈ 2. 034 radians. 6. cos−1 1/ 3 54. 73560
7. 71. 06820, 60. 87840, 35. 79580
8. −2911 9. 4569 10. 7837 11. 24 12. ‖u‖ 29, ‖v‖ 13, and ‖4u 9v‖ 2305 13. 6x − 5y 2z −15. 14. 2x 5y − 9z 40 15. Take the dot product with both u and v. 16. a. 13, 3, 6 b. 5x 13y z 110 17. x y − z 10; they intersect at 2, 5,−3.
4 Selected Answers to the Exercises
18. ⟨x, y, z ⟨5,−3, 7 t⟨9, 22, 17; they intersect at 97 14
, 12 7
, 149 14
.
19. c. 7x 5y − 3z 50 20. c. 7x 11y − 13z 46 and 7x 11y − 13z 104 21. ⟨3,−5, 2 ∘ ⟨2, 4, 7 0; ⟨x, y, z ⟨ 9
22 ,− 21
22 , 0 t⟨−43,−17, 22;
22. 4x y − z 20 23. b. 15x 13y 10z 68; c. ⟨x, y, z ⟨3, 1, 1 t⟨2, 0,−3 24. The direction vector of L is a multiple of the normal vector to . 25. 8x 5y − 4z 2; they intersect at 118
105 , 62
21 , 571
105 26. ⟨x, y, z ⟨5,−2, 1 t⟨3, 7,−4; they intersect at 397
74 ,− 85
74 , 19
37 27. b. x 2z 12. 28. False: the converse is True, but the forward implication is False; u ∘ v 0 means the two
vectors are orthogonal to each other without one of them necessarily being 0 n.
1.4 Exercises
1. ⟨−3, 2, 6; all variables are leading 2. ⟨−9, 4, 0; all variables are leading 3. ⟨−3 − 7r, 2 4r, r, x3 r is free 4. ⟨6 3r, r,−7; x2 r is free 5. ⟨2,−5, r; x3 r is free 6. ⟨8 5r − 2s, r, s; x2 r and x3 s are free 7. ⟨3 5r,−4r,−2 7r, r; x4 r is free 8. ⟨5 − 3r, 6 2r, r,−4; x3 r is free 9. no solutions 10. ⟨5 4r, r,−3 − s, s; x2 r and x4 s are free 11. ⟨7 2r − 6s, r, s,−2; x2 r and x3 s are free 12.
5 3 2
3 r,− 7
3 − 4
3 r, 2
3 − 1
3 r, r ; x4 r is free
13. ⟨−5, 3, 2; all variables are leading 14. ⟨2 − 3r,−4 5r, r; x3 r is free 15. ⟨7 6r, r,−2; x2 r is free 16. ⟨5, 6,−4, 0; all variables are leading 17. ⟨4r, 3 − 7r,−8 − 3r, r; x4 r is free 18. ⟨1 6r, 5 − 4r, r,−4; x3 r is free 19. ⟨−2 5r, r, 3, 7; x2 r is free 20. ⟨−8 3r − 2s,−5 − 4r 6s, r, s; x3 r and x4 s
are free 21. ⟨−2 5r 9s, r,−6 − 4s, s; x2 r and x4 s are free 22. ⟨−5 − 7r − 5s, 2 4r − 3s, 4 − 6r 2s, r, s; x4 r and x5 s are free 23. ⟨5 − 3r 4s 6t,−1 2r 9s − 8t, r, s, t; x3 r, x4 s and x5 t are free. 24. ⟨−5 − 6r, 2 3r, 4 − 2r,−1 − 8r, r; x5 r is free 25. ⟨5 − 3r, 6 2r, r,−4, 9; x3 r is
free 26. ⟨2 − 6r − 3s, r, 7 8s, s,−3; x2 r and x4 s are free 27. ⟨−2 5r − 4s, r, 9 − 7s, 6 − 3s, s; x2 r and x5 s are free 28. no solutions 29. ⟨r, 2 3s, s,−7, 4; x1 r and x3 s are free 30. ⟨4 5r − 3s, 5 − 3r,−2 2r − 4s, 3 − 7r 6s, r, s; x5 r and x6 s are free 31. ⟨7 9r − 4s,−3r s, r,−1 − 6s, 2 − 5s, s; x3 r and x6 s are free 32. ⟨2 − 6r − 3s − 5t, r, 9 8s 2t, s, t,−1, x2 r, x4 s and x5 t are free 33. ⟨3 − 5r,−7 2r, r, 9, 4, x3 r is free 34. ⟨−2 4r − 7s, 5 − 6r 3s, r, 6 − 9s, s,
x3 r, x5 s are free 35. ⟨−2 8r s, 6 − 5r − 7s, r, 3 4s, 8 − 9s, s, x3 r, x6 s are free 36. ⟨−5 − 6r, 2 7r, 3 − 4r, r,−8, 9, x4 r is free 37. Yes, b 3v1 − 5v2 (only solution) 38. b is not in SpanS. 39. Yes. b 3v1 − 2v2 v3 (only solution) 40. Yes. b 1
2 v1 32 v2 (there are infinitely many solutions) 41. b
is not in SpanS.
Selected Answers to the Exercises 5
42. Yes. b 5v1 − 2v2 4v3 (only solution) 43. Yes. b −17v1 13v2 (there are infinitely many solutions)
44. Yes. b 3v1 − 2v2 5v3 (there are infinitely many solutions) 45. Yes. b −2v1 5v2 (there are infinitely many solutions) 46. Yes. b 2v1 − 7v2 3v4 (there are infinitely many solutions) 47. Yes.
b v1 − v2 2v3 (only solution) 48. Yes. b 5v1 − 4v2 (there are infinitely many solutions) 49. 0, 27
,− 3 7
50. 43 11
,− 8 11
,− 8 11
, 2 11
51. − 7 5
,− 8 5
,− 8 5
,− 7 5
52. −2s, 6s − 47 3
, 8 3
, s , where
x4 s ∈ . 53. ⟨−7,−1,−26, 31, 2, 7 54. 8 − 6r − 17t
4 , r,−7,−2 − t
4 , t,−1 , where x5 t ∈ .
55. 8 − 9s,− 1 4 25
4 s, 5 − 5s
4 , 3 4s, 8 − 9s, s , where x6 s ∈ . 56. ⟨5,−3,−9 57.
⟨11,−3, 4 58. ⟨−14,−2, 3, 2 59. ⟨−3 − 3r 4s, r,−2 − 2s, s, 2, x2 r ∈ , x4 s ∈ are free. 60. ⟨3 5r, r,−2, 4, y r ∈ is free. 61. ⟨3 − 5r,−7 2r, r, 4, z r ∈ is free. 62. No solutions. 63. One possible answer: ⟨x, y, z ⟨40, 22, 0 t⟨−43,−25, 2. 64. $1.50 per shirt, $5 per pair of slacks, and $7 per jacket. 65. 1 kilogram of Barley, 3 kilograms of Oats, and 2 kilogram of Soy.
66. The rref is 1 0 − 4
5 159 5
0 1 9 5
331 5
, so d 159 4p/5 and n 331 − 9p/5.
The solution with the smallest number of pennies has p 4, n 59, and d 35. (Note: since we want n ≥ 0, we need p ≤ 36) The solution with the largest number of pennies has p 34, n 5 and d 59.
1.5 Exercises
1. a. consistent, and b. square 2. a. consistent, and b. overdetermined 3. a. inconsistent, and b. overdetermined 4. a. consistent, and b. underdetermined 5. a. inconsistent, and b. underdetermined 6. a. consistent, and b. square 7. a. consistent, and b. square 8. a. consistent, and b. underdetermined. 9. a. consistent, and b. square. 10. a. inconsistent, and b. overdetermined. 11. independent 12. independent 13. dependent 14. dependent 15. dependent 16. independent 17. dependent 18. dependent: 2v1 − v2 v3 03.
6 Selected Answers to the Exercises
19. independent 20. independent 21. dependent: 2v1 − v2 5v3 04. 22. dependent: −4v1 − 7v2 v3 04. 23. dependent: −3v1 − v2 5v3 05. 24. dependent: −2v1 − 3v2 4v3 v4 05 25. a. − 2v1 − 3v2 v3 04 b. 5v1 7v2 v4 04 c. − v2 5v3 2v4 04 26. a. − 2v1 v2 v3 05 b. − 3v1 − 2v2 v4 05 c. − 7v2 − 3v3 2v4 05 27. a. − 3v1 − 5v2 6v3 v4 04 b. − 2v1 − 3v2 5v3 v5 04
c. − v1 7v3 − 3v4 5v5 04 28. a. − 4v1 − 5v2 v3 04 b. − 3v1 − 2v2 v4 2v5 04
c. 7v1 2v3 − 5v4 − 10v5 04 29. a. 5v1 2v2 05 b. 5v1 − 6v3 2v5 05 c. v3 v4 v5 05 30. dependent: 5 vectors in 4 must be dependent. 31. One possible dependence equation is: 32u v − 14u 5v− 4w − 2u − v 2w 0 n. 32. The system will have no solution if r −4 and s ≠ 7
2 . The system will have exactly one
solution if r ≠ −4 and s is any real number. The system will have an infinite number of solutions if r −4 and s 7
2 .
33. In all cases, x is a leading variable. The system will have no solution if s −8 and t ≠ 4. The system will have exactly one solution if s ≠ −8, t is any real number, and r ≠ −6. The system will have an infinite number of solutions involving exactly one free variable in two ways. First, if s −8, t 4, and r ≠ −6, then y is a leading variable and z is a free variable. If r −6, then z is automatically a leading variable because of the 2nd equation, and z − 13
10 . This will satisfy the 3rd equation if and only if 8 s − 13
10 t − 4, so
10t 13s −144. Thus, the second way is to have r −6 and s and t any two real numbers satisfying 10t 13s −144. In this case, y is a free variable. The system will never have an infinite number of solutions involving exactly two free variables.
34. c 22 46. a. False. b. False. c. True. d. False e. True. f. False. g. True. h. False. i. True. j.
False.
1.6 Exercises
1. The corresponding pairs of vectors are parallel to each other. 2. If we denote by S v1, v2 and S/ w 1, w 2, w 3, then we will get:
v1 35 w 1 15
w 2, v2 15 w 1 − 35
w 2, w 1 32 v1 12
v2, w 2 12 v1 − 32
v2,
w 3 2v1 − v2. 3. We should apply the Equality of Spans Theorem; if S v1, v2 and S/ w 1, w 2, then
we will get: v1 13
w 1 23 w 2, v2 53
w 1 163 w 2, w 1 8v1 − v2, w 2 − 52
v1 12 v2.
4. Although both Theorems are applicable, the first Theorem will certainly be easier to apply: corresponding pairs of vectors are parallel to each other.
5. a. S consists of 6 vectors from 3, so S is certainly dependent. b. v2 and v4 are parallel
Selected Answers to the Exercises 7
to v1. c. Eliminate v2 and v4, to get: S / v1, v3, v5, v6 . You could also eliminate v1 and v2 and keep v4, or eliminate the v1 and v4 and keep v2. d. v5 53
v1 2v3. e.
Eliminate either v1 or v3 or v5 to get a set with 3 vectors left. One possible answer is S // v1, v3, v6 . f. The rref of the 3 3 matrix you obtained should not have any free variables.
6. a. S consists of 5 vectors from 4, so S is certainly dependent. b. v4 is parallel to v2. c. Eliminate either v2 or v4, so one possible answer is: S / v1, v2, v3, v5 d. v3 3v1 − 2v2 and v5 2v1 v2. e. two vectors are left; one possible answer is: S // v1, v2 . f. the two vectors (no matter which you picked) are obviously not parallel.
7. a, b and d only. 8. a, d and e only. 9. a, b, c, d and f only. 10. S / v1, v2, v3 ; v4 3v1 2v2 − 4v3. 11. S / v1, v2 ; v3 3v1 − 2v2 ; v4 2v1 3v2. 12. S / v1, v3 ; v2 −5v1; v4 3v1 5v3. 13. S / v1, v2, v3 ; v4 3v1 4v2 − 2v3 ; v5 2v1 3v2 − v3 . 14. S / v1, v2, v4 ; v3 4v1 7v2; v5 3v1 4v2 − 2v4 . 15. S / v1, v3, v4 ; v2 −4v1; v5 12
v1 − 32 v3 12
v4 .
16. S / v1, v3, v5 ; v2 3v1; v4 4v1 2v3. 17. S / v1, v2 ; v3 4v1 3v2; v4 −v1 2v2; v5 −2v1 − v2. 18. S / v1, v2 ; v3 2v1 − 3v2. 19. S / v1, v2, v3 . 20. S / v1, v2 ; v3 v1 2v2; v4 −6v1 5v2. 21. S / v1, v2, v4 ; v3 5v1 7v2. 22. S / v1, v3 ; v2 −3v1; v4 5v1 4v3. 23. S / v1, v2, v4 ; v3 52
v1 92 v2; v5 v1 7v2 5v4 .
24. S / v1, v2, v3, v5; v4 5v1 4v2 − 2v3. 25. S / v1, v2, v3 ; v4 5v1 4v2 − 2v3; v5 7v1 5v2 − 4v3. 26. S / v1, v2 ; v3 17
v1 − 57 v2.
27. S / v1, v2, v3 . 28. S / v1, v2 ; v3 47
v1 297 v2; v4 17
v1 − 57 v2.
29. S / v1, v2, v4 ; v3 5v1 8v2. 30. S / v1, v2, v3 ; v4 2v1 − 3v2 − 4v3. 31. S / v1, v3, v5 ; v2 16
v1; v4 76 v1 − 9v3.
32. S / v1, v2, v4 ; v3 −6v1 5v2; v5 5v1 − 3v2. 33. S / v1, v2, v4, v5 ; v3 5v1 8v2. 34. S / v1, v2, v4 ; v3 7v1 − 9v2 ; v5 2v1 v2 5v4; v6 4v1 − 6v2 − 3v4. 35. S / v1, v3, v6 ; v2 −4v1; v4 5/3v1; v5 5/3v1 2v3. 36. S / v1, v2; v3 3v1 − 2v2; v4 −5v2; v5 2v1 v2. 37. S / v1, v2; v3 −2/3v1 7/3v2; v4 1/3v1 1/3v2;
v5 −1/3v1 2/3v2.
8 Selected Answers to the Exercises
38. S / v1, v2, v4 ; v3 − 32 v2; v5 12
v2 v4.
39. S / v1, v2, v4 ; v3 2v1 − 4v2; v5 2v1 − 3v2. 40. S / v1, v2, v4, v5 ; v3 7v1 − 4v2; v6 6v1 − 7v2 3v4 − 5v5. 41. S / v1, v2, v4 ; v3 5v1 8v2; v5 5v1 7v2 4v4; v6 4v1 3v2 2v4. 42. S / v1, v2; v3 −2v1; v4 −v1 v2; v5 −2v1 5v2. 43. S / v1, v2, v4 ; v3 4v1 3v2; v5 2v1 5v2 3v4 44. S / v1, v2, v3, v5 ; v4 3v1 4v2 − 2v3; v6 4v1 2v2 − 3v3 − 5v5. 45. S / v1, v2, v3, v5, v6 ; v4 5v1 3v2 − 2v3. 46. a. Two non-parallel vectors are independent. b. x 2z 0. c. only e2 is in SpanS
d. Yes, because e1 ∉ SpanS. d. No, because e2 ∈ SpanS. e. Yes, because e3 ∉ SpanS.
47. b. Yes. c. No. d. No. e. Yes. 48. b.
1 0 0 0 −2 0 0 1 −1
0 1 0 0 5 2
0 0 − 37 26
27 26
0 0 1 0 1 2
0 0 − 7 26
3 26
0 0 0 1 0 0 0 8 13
17 13
0 0 0 0 0 1 0 6 13
16 13
0 0 0 0 0 0 1 7 13
− 16 13
c. independent d. dependent e. independent f. independent g. independent 53. a. False. b. True. c. False. d. True. e. False. f. False. g. False. h. False. i. False.
1.7 Exercises
1. ⟨7, 5. 2. It doesn’t contain the origin. 3. ⟨7, 3, 0,⟨0, 4, 7 is one possibility (you can also use ⟨4, 0,−3 as a second vector. 4. ⟨5, 0, 2,⟨0, 1, 0 5. It doesn’t contain the origin. 6. v1, v2, v4 ; dimW 3 7. v1, v2, v3 ; dimW 3 8. v1, v3, v6 ; dimW 3 9. v1, v2, v4 ; dimW 3 10. v1, v2, v4, v5 ; dimW 4 11. v1, v2, v4 ; dimW 3 12. v1, v3, v6 ; dimW 3 13. v1, v2; dimW 2 14. ⟨5,−3, 6, 7, ⟨3,−1, 4, 5; dimW 2 15. ⟨5,−3, 6, 7, ⟨3,−1, 4, 5, ⟨5, 1, 8,−3; dimW 3 16. ⟨5,−3, 6, 7, ⟨3,−1, 4, 5, ⟨1, 3,−1, 1; dimW 3 17. ⟨7, 5,−4, 3, 9, ⟨4, 3,−2, 1, 5; dimW 2 18. ⟨7, 5,−4, 3, 9, ⟨4, 3,−2, 1, 5, ⟨4, 3,−5, 9, 5; dimW 3 19. ⟨7, 5,−4, 3, 9, ⟨4, 3,−2, 1, 5, ⟨4, 3,−5, 4, 5; dimW 3 20. ⟨5,−3, 7,−4, 6, 3, ⟨9,−7, 8,−9, 4, 7, ⟨4,−5,−3,−6,−7, 5; dimW 3 21. ⟨7,−3, 4, 2,−5, 2, ⟨5,−2, 3, 3,−4, 1, ⟨−4, 1,−3,−8, 5, 1; dimW 3
Selected Answers to the Exercises 9
22. ⟨7,−3, 4, 2,−5, 2, ⟨5,−2, 3, 3,−4, 1, ⟨6,−4, 3,−9,−2, 5, ⟨−4, 1,−3,−8, 5, 1; dimW 4
23. ⟨7,−3, 4, 2,−5, 2, ⟨5,−2, 3, 3,−4, 1, ⟨−4, 1,−3,−2, 4,−1; dimW 3 24. ⟨7,−3, 4, 2,−5, 2, ⟨5,−2, 3, 3,−4, 1, ⟨8,−4, 3,−9,−2, 5, ⟨−4, 1,−3,−2, 4,−1;
dimW 4 25. ⟨7,−3, 4, 2,−5, 2, ⟨5,−2, 3, 3,−4, 1, ⟨5,−3, 2,−8,−1, 4, ⟨8,−4, 3,−9,−2, 5, ⟨−4, 1,−3,−2, 4
dimW 5 26. the xz-plane. ⟨1, 0, 0, ⟨0, 0, 1; dimW 2 27. the x-axis. ⟨1, 0, 0; dimW 1 28. W is not a subspace. It is not closed under addition. 29. ⟨5, 0, 1, 0, ⟨0,−1, 0, 1; dimW 2 30. ⟨0, 5, 1, 0, 0, ⟨0, 6, 0, 1, 0, ⟨−7, 0, 0, 0, 1; dimW 3 31. It does not contain the origin. 32. W is not a subspace, because it is not closed under scalar multiplication.
1.8 Exercises
1. rowspaceA: ⟨1, 0, 0, 3,⟨0, 1, 0, 2,⟨0, 0, 1,−4; colspaceA: ⟨2,−3, 4,⟨−3, 0,−5,⟨3,−1,−2; nullspaceA: ⟨−3,−2, 4, 1; nullspaceA 03 ; rankA 3 rankA; nullityA 1; nullityA 1; 3 1 4 and 3 0 3; ⟨2,−3, 3,−12 2⟨1, 0, 0, 3 − 3⟨0, 1, 0, 2 3⟨0, 0, 1,−4 ⟨−3, 0,−1,−5 −3⟨1, 0, 0, 3 − ⟨0, 0, 1,−4; ⟨4,−5,−2, 10 4⟨1, 0, 0, 3 − 5⟨0, 1, 0, 2 − 2⟨0, 0, 1,−4
2. rowspaceA: ⟨1,−5, 0, 3,⟨0, 0, 1, 5; colspaceA: ⟨−2, 4,−3,⟨3,−2, 4; nullspaceA: ⟨5, 1, 0, 0,⟨−3, 0,−5, 1; nullspaceA: ⟨−10, 1, 8; rankA 2 rankA; nullityA 2; nullityA 1; 2 2 4 and 2 1 3; ⟨−2, 10, 3, 9 −2⟨1,−5, 0, 3 3⟨0, 0, 1, 5 ⟨4,−20,−2, 2 −4⟨1,−5, 0, 3 − 2⟨0, 0, 1, 5; ⟨−3, 15, 4, 11 −3⟨1,−5, 0, 3 4⟨0, 0, 1, 5
3. rowspaceA: ⟨1, 0, 4, 0, 3,⟨0, 1, 7, 0, 4,⟨0, 0, 0, 1,−2; colspaceA: ⟨5,−2, 3,⟨−2, 3,−4,⟨−1,−3, 2; nullspaceA: ⟨−4,−7, 1, 0, 0,⟨−3,−4, 0, 2, 1; nullspaceA 03 ; rankA 3 rankA; nullityA 2; nullityA 0; 3 2 5 and 3 0 3; ⟨5,−2, 6,−1, 9 5⟨1, 0, 4, 0, 3 − 2⟨0, 1, 7, 0, 4 − ⟨0, 0, 0, 1,−2 ⟨−2, 3, 13,−3, 12 −2⟨1, 0, 4, 0, 3 3⟨0, 1, 7, 0, 4 − 3⟨0, 0, 0, 1,−2 ⟨3,−4,−16, 2,−11 3⟨1, 0, 4, 0, 3 − 4⟨0, 1, 7, 0, 4 2⟨0, 0, 0, 1,−2
4. rowspaceA: ⟨1, 3, 0, 4, 0,⟨0, 0, 1, 2, 0,⟨0, 0, 0, 0, 1; colspaceA: ⟨−1,−3, 2,⟨−2, 3,−4,⟨5,−2, 3; nullspaceA: ⟨−3, 1, 0, 0, 0,⟨−4, 0,−2, 1, 0; nullspaceA 03 ; rankA 3 rankA; nullityA 2; nullityA 0; 3 2 5 and 3 0 3; ⟨−1,−3,−2,−8, 5 −⟨1, 3, 0, 4, 0 − 2⟨0, 0, 1, 2, 0 5⟨0, 0, 0, 0, 1 ⟨−3,−9, 3,−6,−2 −3⟨1, 3, 0, 4, 0 3⟨0, 0, 1, 2, 0 − 2⟨0, 0, 0, 0, 1
10 Selected Answers to the Exercises
⟨2, 6,−4, 0, 3 2⟨1, 3, 0, 4, 0 − 4⟨0, 0, 1, 2, 0 3⟨0, 0, 0, 0, 1 5. rowspaceA: ⟨1, 0, 4,−1,−2,⟨0, 1, 3, 2,−1; colspaceA: ⟨−2, 3,−5,⟨5,−2, 3;
nullspaceA: ⟨−4,−3, 1, 0, 0,⟨1,−2, 0, 1, 0,⟨2, 1, 0, 0, 1; nullspaceA: ⟨1, 19, 11; rankA 2 rankA; nullityA 3; nullityA 1; 2 3 5 and 2 1 3; ⟨−2, 5, 7, 12,−1 −2⟨1, 0, 4,−1,−2 5⟨0, 1, 3, 2,−1; ⟨3,−2, 6,−7,−4 3⟨1, 0, 4,−1,−2 − 2⟨0, 1, 3, 2,−1 ⟨−5, 3,−11, 11, 7 −5⟨1, 0, 4,−1,−2 3⟨0, 1, 3, 2,−1
6. rowspaceA: ⟨1, 0, 0,⟨0, 1, 0,⟨0, 0, 1; colspaceA: ⟨3, 7, 1,−9,⟨−2,−4, 0,−5,⟨5,−6, 8, 2; nullspaceA 03 ; nullspaceA: ⟨−14, 243, 141, 200; rankA 3 rankA; nullityA 0; nullityA 1; 2 2 4 and 2 2 4; ⟨3,−2, 5 3⟨1, 0, 0 − 2⟨0, 1, 0 5⟨0, 0, 1; ⟨7, 4,−6 7⟨1, 0, 0 4⟨0, 1, 0 − 6⟨0, 0, 1 ⟨1, 0, 8 1⟨1, 0, 0 8⟨0, 0, 1; ⟨−9,−5, 2 −9⟨1, 0, 0 − 5⟨0, 1, 0 2⟨0, 0, 1
7. rowspaceA: ⟨1, 0, 1,−6,⟨0, 1, 2, 5; colspaceA: ⟨2, 1,−2,−2,⟨3,−2, 1,−4; nullspaceA: ⟨−1,−2, 1, 0,⟨6,−5, 0, 1; nullspaceA: ⟨3, 8, 7, 0,⟨8,−2, 0, 7; rankA 2 rankA; nullityA 2; nullityA 2; 2 2 4 and 2 2 4; ⟨2, 3, 8, 3 2⟨1, 0, 1,−6 3⟨0, 1, 2, 5; ⟨1,−2,−3,−16 ⟨1, 0, 1,−6 − 2⟨0, 1, 2, 5 ⟨−2, 1, 0, 17 −2⟨1, 0, 1,−6 ⟨0, 1, 2, 5; ⟨−2,−4,−10,−8 −2⟨1, 0, 1,−6 − 4⟨0, 1, 2, 5
8. rowspaceA: ⟨1,−3, 0, 5,⟨0, 0, 1, 4; colspaceA: ⟨−3, 7, 5, 4,⟨1,−4, 2,−3; nullspaceA: ⟨3, 1, 0, 0,⟨−5, 0,−4, 1; nullspaceA: ⟨34, 11, 5, 0,⟨−1,−1, 0, 1; rankA 2 rankA; nullityA 2; nullityA 2; 2 2 4 and 2 2 4; ⟨−3, 9, 1,−11 −3⟨1,−3, 0, 5 ⟨0, 0, 1, 4; ⟨7,−21,−4, 19 7⟨1,−3, 0, 5 − 4⟨0, 0, 1, 4 ⟨5,−15, 2, 33 5⟨1,−3, 0, 5 2⟨0, 0, 1, 4; ⟨4,−12,−3, 8 4⟨1,−3, 0, 5 − 3⟨0, 0, 1, 4
9. rowspaceA: ⟨2, 0, 5, 0, 2,⟨0, 2, 9, 0, 14,⟨0, 0, 0, 1, 5; colspaceA: ⟨0,−7, 8,−2,⟨2, 1,−2,−2,⟨−4, 3,−1, 6; nullspaceA: ⟨−5,−9, 2, 0, 0,⟨−1,−7, 0,−5, 1; nullspaceA: ⟨4,−6,−4, 5; rankA 3 rankA; nullityA 2; nullityA 1; 3 2 5 and 3 1 4; ⟨0, 2, 9,−4,−6 ⟨0, 2, 9, 0, 14 − 4⟨0, 0, 0, 1, 5; ⟨−7, 1,−13, 3, 15 − 7
2 ⟨2, 0, 5, 0, 2 1
2 ⟨0, 2, 9, 0, 14 3⟨0, 0, 0, 1, 5
⟨8,−2, 11,−1,−11 4⟨2, 0, 5, 0, 2 − ⟨0, 2, 9, 0, 14 − ⟨0, 0, 0, 1, 5 ⟨−2,−2,−14, 6, 14 −⟨2, 0, 5, 0, 2 − ⟨0, 2, 9, 0, 14 6⟨0, 0, 0, 1, 5
10. rowspaceA: ⟨1, 0, 0, 5, 7,⟨0, 1, 0, 4, 5,⟨0, 0, 1,−2,−4; colspaceA: ⟨3, 7, 1,−9,⟨−2,−4, 0, 6,⟨5, 6, 3,−9; nullspaceA: ⟨−5,−4, 2, 1, 0,⟨−7,−5, 4, 0, 1; nullspaceA: ⟨9, 6,−6, 7; rankA 3 rankA; nullityA 2; nullityA 1; 3 2 5 and 3 1 4; ⟨3,−2, 5,−3,−9 3⟨1, 0, 0, 5, 7 − 2⟨0, 1, 0, 4, 5 5⟨0, 0, 1,−2,−4 ⟨7,−4, 6, 7, 5 7⟨1, 0, 0, 5, 7 − 4⟨0, 1, 0, 4, 5 6⟨0, 0, 1,−2,−4 ⟨1, 0, 3,−1,−5 ⟨1, 0, 0, 5, 7 3⟨0, 0, 1,−2,−4; ⟨−9, 6,−9,−3, 3 −9⟨1, 0, 0, 5, 7 6⟨0, 1, 0, 4, 5 − 9⟨0, 0, 1,−2,−4
11. rowspaceA: ⟨7, 0, 4, 1,⟨0, 7, 29,−5; colspaceA: ⟨15,−3, 13,−9,−11,⟨3,−2, 4, 1, 2; nullspaceA: ⟨−4,−29, 7, 0,⟨−1, 5, 0, 7; nullspaceA: ⟨−2, 3, 3, 0, 0,⟨1, 2, 0, 1, 0,⟨4, 9, 0, 0, 3; rankA 2 rankA; nullityA 2; nullityA 3; 2 2 4 and 2 3 5;
Selected Answers to the Exercises 11
⟨15, 3, 21, 0 15 7 ⟨7, 0, 4, 1 3
7 ⟨0, 7, 29,−5;
⟨−3,−2,−10, 1 −3 7 ⟨7, 0, 4, 1 − 2
7 ⟨0, 7, 29,−5
⟨13, 4, 24,−1 13 7 ⟨7, 0, 4, 1 4
7 ⟨0, 7, 29,−5;
⟨−9, 1,−1,−2 −9 7 ⟨7, 0, 4, 1 1
7 ⟨0, 7, 29,−5
⟨−11, 2, 2,−3 −11 7
⟨7, 0, 4, 1 2 7 ⟨0, 7, 29,−5
12. rowspaceA: ⟨1, 0, 5, 0,⟨0, 1, 8, 0,⟨0, 0, 0, 1; colspaceA: ⟨3,−2,−1, 2,⟨7,−4, 3, 6,⟨1, 0, 5, 1; nullspaceA: ⟨−5,−8, 1, 0; nullspaceA: ⟨−1, 2,−2, 1, 0,⟨23,−13, 14, 0, 2; rankA 3 rankA; nullityA 1; nullityA 2; 3 1 4 and 3 2 5; ⟨3,−2,−1, 2 3⟨1, 0, 5, 0 − 2⟨0, 1, 8, 0 2⟨0, 0, 0, 1; ⟨7,−4, 3, 6 7⟨1, 0, 5, 0 − 4⟨0, 1, 8, 0 6⟨0, 0, 0, 1 ⟨1, 0, 5, 1 ⟨1, 0, 5, 0 ⟨0, 0, 0, 1; ⟨−9, 6, 3,−8 −9⟨1, 0, 5, 0 6⟨0, 1, 8, 0 − 8⟨0, 0, 0, 1 ⟨4,−3,−4, 9 4⟨1, 0, 5, 0 − 3⟨0, 1, 8, 0 9⟨0, 0, 0, 1
13. rowspaceA: ⟨1, 0, 0, 2,⟨0, 1, 0,−3,⟨0, 0, 1,−4; colspaceA: ⟨5,−3, 3,−9,−1,⟨3,−2, 4, 1, 2,⟨2,−1, 3,−1, 2; nullspaceA: ⟨−2, 3, 4, 1; nullspaceA: ⟨30, 29,−9, 4,⟨2,−3,−5, 0, 4; rankA 3 rankA; nullityA 1; nullityA 2; 3 1 4 and 3 2 5; ⟨5, 3, 2,−7 5⟨1, 0, 0, 2 3⟨0, 1, 0,−3 2⟨0, 0, 1,−4; ⟨−3,−2,−1, 4 −3⟨1, 0, 0, 2 − 2⟨0, 1, 0,−3 − ⟨0, 0, 1,−4 ⟨3, 4, 3,−18 3⟨1, 0, 0, 2 4⟨0, 1, 0,−3 3⟨0, 0, 1,−4; ⟨−9, 1,−1,−17 −9⟨1, 0, 0, 2 ⟨0, 1, 0,−3 − ⟨0, 0, 1,−4 ⟨−1, 2, 2,−16 −1⟨1, 0, 0, 2 2⟨0, 1, 0,−3 2⟨0, 0, 1,−4
14. rowspaceA: ⟨6, 1, 0, 7, 0,⟨0, 0, 1,−9, 0,⟨0, 0, 0, 0, 1; colspaceA: ⟨12,−6, 18,−6, 12,⟨3,−2, 4, 1, 2,⟨5,−3, 0, 7,−1; nullspaceA: ⟨−1, 6, 0, 0, 0,⟨−7, 0, 54, 6, 0; nullspaceA Span⟨1, 4, 1, 1, 0,⟨−4,−9,−5, 0, 7; rankA 3 rankA; nullityA 2; nullityA 2; 3 2 5 and 3 2 5; ⟨12, 2, 3,−13, 5 2⟨6, 1, 0, 7, 0 3⟨0, 0, 1,−9, 0 5⟨0, 0, 0, 0, 1 ⟨−6,−1,−2, 11,−3 −⟨6, 1, 0, 7, 0 − 2⟨0, 0, 1,−9, 0 − 3⟨0, 0, 0, 0, 1 ⟨18, 3, 4,−15, 0 3⟨6, 1, 0, 7, 0 4⟨0, 0, 1,−9, 0; ⟨−6,−1, 1,−16, 7 −6⟨6, 1, 0, 7, 0 ⟨0, 0, 1,−9, 0 7⟨0, 0, 0, 0, 1 ⟨12, 2, 2,−4,−1 2 1, 1
6 , 0, 7
6 , 0 2⟨0, 0, 1,−9, 0 − ⟨0, 0, 0, 0, 1
15. rowspaceA: ⟨1, 0, 5, 0, 0,⟨0, 1, 8, 0, 0,⟨0, 0, 0, 1, 0,⟨0, 0, 0, 0, 1; colspaceA: ⟨3, 7, 1,−9, 4,⟨−2,−4, 0, 6,−3,⟨2, 6, 1,−8, 9,⟨5, 6, 3,−9, 7; nullspaceA: ⟨−5,−8, 1, 0, 0; nullspaceA: ⟨91, 82,−74, 93, 16; rankA 4 rankA; nullityA 1 nullityA; 4 1 5 for both matrices; ⟨3,−2,−1, 2, 5 3⟨1, 0, 5, 0, 0 − 2⟨0, 1, 8, 0, 0 2⟨0, 0, 0, 1, 0 5⟨0, 0, 0, 0, 1 ⟨7,−4, 3, 6, 6 7⟨1, 0, 5, 0, 0 − 4⟨0, 1, 8, 0, 0 6⟨0, 0, 0, 1, 0 6⟨0, 0, 0, 0, 1 ⟨1, 0, 5, 1, 3 ⟨1, 0, 5, 0, 0 5⟨0, 0, 0, 1, 0 3⟨0, 0, 0, 0, 1 ⟨−9, 6, 3,−8,−9 9⟨1, 0, 5, 0, 0 − 6⟨0, 1, 8, 0, 0 − 8⟨0, 0, 0, 1, 0 − 9⟨0, 0, 0, 0, 1 ⟨4,−3,−4, 9, 7 4⟨1, 0, 5, 0, 0 − 3⟨0, 1, 8, 0, 0 9⟨0, 0, 0, 1, 0 7⟨0, 0, 0, 0, 1
16. rowspaceA: ⟨1, 0, 7, 0, 2, 4,⟨0, 1,−9, 0, 1,−6,⟨0, 0, 0, 1, 5,−3; colspaceA: ⟨2,−1, 3,−1, 2,⟨3,−2, 4, 1, 2,⟨1,−3, 2,−2,−1;
12 Selected Answers to the Exercises
nullspaceA: ⟨−7, 9, 1, 0, 0, 0,⟨−2,−1, 0,−5, 1, 0,⟨−4, 6, 0, 3, 0, 1; nullspaceA: ⟨−19, 1, 14, 3, 0,⟨0,−1,−1, 0, 1; rankA 3 rankA; nullityA 3; nullityA 2; 3 3 6 and 3 2 5; ⟨2, 3,−13, 1, 12,−13 2⟨1, 0, 7, 0, 2, 4 3⟨0, 1,−9, 0, 1,−6 ⟨0, 0, 0, 1, 5,−3 ⟨−1,−2, 11,−3,−19, 17 −1⟨1, 0, 7, 0, 2, 4 − 2⟨0, 1,−9, 0, 1,−6 − 3⟨0, 0, 0, 1, 5,−3 ⟨3, 4,−15, 2, 20,−18 3⟨1, 0, 7, 0, 2, 4 4⟨0, 1,−9, 0, 1,−6 2⟨0, 0, 0, 1, 5,−3 ⟨−1, 1,−16,−2,−11,−4 −⟨1, 0, 7, 0, 2, 4 ⟨0, 1,−9, 0, 1,−6 − 2⟨0, 0, 0, 1, 5,−3 ⟨2, 2,−4,−1, 1,−1 2⟨1, 0, 7, 0, 2, 4 ⟨20, 1,−9, 0, 1,−6 − ⟨0, 0, 0, 1, 5,−3
17. ⟨3, 0,−2, 0 x4⟨5,−4, 7, 1 18. 5, 6, 0, − 4 x3⟨−3, 2, 1, 0 19. 5, 0,−3, 0 x2⟨4, 1, 0, 0 x4 0, 0,−1, 1 20. ⟨7, 0, 0,−2 x2⟨2, 1, 0, 0 x3⟨−6, 0, 1, 0 21. ⟨2,−4, 0 x3⟨−3, 5, 1 22. ⟨7, 0,−2 x2⟨6, 1, 0 23. ⟨0, 3,−8, 0 x4⟨4,−7,−3, 1 24. ⟨−2, 0, 3, 7 x2⟨5, 1, 0, 0 25. ⟨−8,−5, 0, 0 x3⟨3,−4, 1, 0 x4⟨−2, 6, 0, 1 26. ⟨−5, 2, 4, 0, 0 x4⟨−7, 4,−6, 1, 0 x5⟨−5, 3, 2, 0, 1 27. ⟨5,−1, 0, 0, 0 x3⟨−3, 2, 1, 0, 0 x4⟨4, 9, 0, 1, 0 x5⟨6,−8, 0, 0, 1 28. 5, 6, 0,−4, 9 x3 −3, 2, 1, 0, 0 29. ⟨−2, 0, 9, 6, 0 x2⟨5, 1, 0, 0, 0 x5⟨−4, 0,−7,−3, 1 30. ⟨0, 2, 0,−7, 4 x1⟨1, 0, 0, 0, 0 x3⟨0, 3, 1, 0, 0 31. ⟨4, 5,−2, 3, 0, 0 x5⟨5,−3, 2,−7, 1, 0 x6⟨−3, 0,−4, 6, 0, 1 32. ⟨2, 0, 9, 0, 0,−1 x2⟨−6, 1, 0, 0, 0, 0 x4⟨−3, 0, 8, 1, 0, 0 x5⟨−5, 0, 2, 0, 1, 0 33. 3,−7, 0, 9, 4 x3 −5, 2, 1, 0, 0 34. −2, 5, 0, 6, 0 x3 4,−6, 1, 0, 0 x5⟨−7, 3, 0,−9, 1 35. −2, 6, 0, 3, 8, 0 x3 8,−5, 1, 0, 0, 0 x6⟨1,−7, 0, 4,−9, 1 36. −5, 2, 3, 0,−8, 9 x4 −6, 7,−4, 1, 0, 0 37. ⟨7,−8, 0 x3⟨4,−5, 1 38. ⟨7, 6,−4, 0 x4 −3, 2, 5, 1 39. ⟨4,−3, 0, 9 x3 2, − 7, 1, 0 40. ⟨3,−2, 0, 0 x2 4, 1, 0, 0 x4 −5, 0, − 7, 1 41. ⟨−3, 2, 0, 0, 0 x3 −4, 7, 1, 0, 0 x4 9,−3, 0, 1, 0 x5 −6, 5, 0, 0, 1 42. ⟨5, 0, 0, 4, 0 x2 −4, 1, 0, 0, 0 x3 6, 0, 1, 0, 0 x5 −7, 0, 0, 3, 1 43. ⟨6, 0,−11, 0, 0 x2 5, 1, 0, 0, 0 x4 −4, 0, 2, 1, 0 x5 7, 0, − 4, 0, 1 44. ⟨−5, 3,−4, 2, 0 x5 −3, 5, 2, − 7, 1 45. 3, 8, 0, − 2, 7, 0 x3 −5, 3, 1, 0, 0, 0 x6 1, 0, 0,−4, 6, 1 46. rowspaceA: ⟨1, 0, 4, 5,⟨0, 1,−2,−3; colspaceA: ⟨3, 5, 16, ⟨2, 7, 29;
nullspaceA: ⟨−4, 2, 1, 0,⟨−5, 3, 0, 1; nullspaceA: ⟨3,−5, 1; rankA 2 rankA; nullityA 2; nullityA 1; 2 2 4 and 2 1 3.
47. rowspaceA: ⟨1, 0, 4, 0, ⟨0, 1,−3, 0, ⟨0, 0, 0, 1; colspaceA: ⟨5,−4, 3, ⟨6,−7, 2, ⟨1, 2, 3; nullspaceA: ⟨−4, 3, 1, 0; nullspaceA 03 ;
Selected Answers to the Exercises 13
rankA 3 rankA; nullityA 1; nullityA 0; 3 1 4; 3 0 3. 48. rowspaceA: ⟨1, 0, 4, 0, 6, ⟨0, 1,−3, 0,−3, ⟨0, 0, 0, 1,−4;
colspaceA: ⟨5, 4, 3, ⟨6, 7, 2, ⟨1,−2, 3; nullspaceA: ⟨−4, 3, 1, 0, 0, ⟨−6, 3, 0, 4, 1; nullspaceA 03 ; rankA 3 rankA; nullityA 2; nullityA 0; 3 2 5; 3 0 3.
49. rowspaceA: ⟨1, 0, 4, 0, 2, ⟨0, 1,−3, 0,−5, ⟨0, 0, 0, 1, 7; colspaceA: ⟨3, 5, 1, 4, ⟨4, 7, 2, 3, ⟨3, 4,−1, 2; nullspaceA: ⟨−4, 3, 1, 0, 0, ⟨−2, 5, 0,−7, 1; nullspaceA: ⟨3,−2, 1, 0; rankA 3 rankA; nullityA 2; nullityA 1; 3 2 5; 3 1 4.
50. rowspaceA: ⟨1, 0, 5, 0,−8, ⟨0, 1,−7, 0, 3, ⟨0, 0, 0, 1, 7; colspaceA: ⟨4, 6, 17, 28, ⟨2, 3, 8, 13, ⟨5, 7, 20, 30; nullspaceA: ⟨−5, 7, 1, 0, 0,⟨8,−3, 0,−7, 1; nullspaceA: ⟨9,−5,−2, 1; rankA 3 rankA; nullityA 2; nullityA 1; 3 2 5; 3 1 4.
51. rowspaceA: ⟨1, 0,−7, 0,−9, ⟨0, 1, 4, 0, 3, ⟨0, 0, 0, 1, 2; colspaceA: ⟨4, 2, 5, 7, 10, ⟨11, 5, 12, 9, 19, ⟨9, 4, 10, 8, 17; nullspaceA: ⟨7,−4, 1, 0, 0, ⟨9,−3, 0,−2, 1; nullspaceA: ⟨6,−3,−5, 1, 0, ⟨3, 4,−6, 0, 1; rankA 3 rankA; nullityA 2 nullityA; 3 2 5 for both A and A.
52. rowspaceA: ⟨1, 0, 4, 0, 0, ⟨0, 1,−5, 0, 0, ⟨0, 0, 0, 1, 0, ⟨0, 0, 0, 0, 1; colspaceA: ⟨3, 1, 0,−1,−4, ⟨2, 1, 2,−6,−7, ⟨−1,−1,−3, 7, 8, ⟨−1, 0, 4,−13,−6; nullspaceA: ⟨−4, 5, 1, 0, 0; nullspaceA: ⟨3,−8, 4, 1, 0; rankA 4 rankA; nullityA 1 nullityA; 4 1 5 for both A and A.
53. rowspaceA: ⟨1, 0,−2,−3, 0, ⟨0, 1, 6, 5, 0, ⟨0, 0, 0, 0, 1; colspaceA: ⟨4, 9, 0, 11,−6,−9, ⟨2, 4, 2, 5,−2,−4, ⟨1, 2, 1, 2, 1, 1; nullspaceA: ⟨2,−6, 1, 0, 0, ⟨3,−5, 0, 1, 0; nullspaceA: ⟨−9, 4, 1, 0, 0, 0, ⟨−5,−2, 0, 4, 1, 0, ⟨−3,−5, 0, 6, 0, 1; rankA 3 rankA; nullityA 2; nullityA 3; 3 2 5, and 3 3 6.
54. rowspaceA: ⟨1, 0, 1, 0, 0, ⟨0, 1,−7, 0, 0, ⟨0, 0, 0, 1, 0, ⟨0, 0, 0, 0, 1; colspaceA: ⟨3, 0, 12,−1, 12,−1,⟨1,−1, 1,−1, 0, 0,⟨−2,−2,−14,−1,−17, 4,⟨0, 5, 15, 4, 22,−6; nullspaceA: ⟨−1, 7, 1, 0, 0; nullspaceA: ⟨−4,−3, 1, 0, 0, 0,⟨−5,−2, 0,−3, 1, 0; rankA 4 rankA; nullityA 1; nullityA 2; 4 1 5, and 4 2 6.
55. rowspaceA: ⟨1, 0, 2, 0, 0, 5,⟨0, 1, 3, 0, 0, 2,⟨0, 0, 0, 1, 0, 7,⟨0, 0, 0, 0, 1, 4; colspaceA: ⟨3, 4, 1,−6,−1, 9, ⟨1,−3,−2, 1, 1, 2, ⟨0, 2, 1,−2, 2,−18, ⟨−4,−5,−1, 9,−3, 18; nullspaceA: ⟨−2,−3, 1, 0, 0, 0, ⟨−5,−2, 0,−7,−4, 1; nullspaceA: ⟨−2, 5,−8, 1, 0, 0, ⟨−4, 3,−2, 0, 7, 1; rankA 4 rankA; nullityA 2 nullityA; 4 2 6 for both A and A.
56. rowspaceA: ⟨1, 0, 4, 5, 0, 0, 3, ⟨0, 1, 9, 8, 0, 0,−4, ⟨0, 0, 0, 0, 1, 0,−4, ⟨0, 0, 0, 0, 0, 1, 5; colspaceA: ⟨3,−4,−3, 1,−5, ⟨−1, 3, 2, 4, 3, ⟨2, 2, 1, 12, 0, ⟨2, 3, 1, 17, 1; nullspaceA: ⟨−4,−9, 1, 0, 0, 0, 0, ⟨−5,−8, 0, 1, 0, 0, 0, ⟨−3, 4, 0, 0, 4,−5, 1; nullspaceA: ⟨−3,−5, 4, 1, 0; rankA 4 rankA; nullityA 3; nullityA 1;
14 Selected Answers to the Exercises
4 3 7, and 4 1 5. 63. 6 13. 67. a. False b. True c. False d. False e. True f. True g. False h. True i.
False j. False k. True l. True m. False n. False o. False.
1.9 Exercises
1. 1 0 43
11
0 1 − 13 11
; W: ⟨11, 0, 43,⟨0, 11,−13; W: ⟨−43, 13, 11; dimW 2;
dimW 1; 2 1 3.
2. 1 0 26
17 − 1
17
0 1 11 34 − 11
17
; W: ⟨17, 0, 26,−1,⟨0, 34, 11,−22; W:
⟨−52,−11, 34, 0,⟨1, 11, 0, 17; dimW 2; dimW 2; 2 2 4.
3. 1 0 5
11 1 11
11 11
0 1 − 20 11 − 37
11 18 11
; W: ⟨11, 0, 5, 1, 12,⟨0, 11,−20,−37, 18;
W: ⟨−5, 20, 11, 0, 0,⟨−1, 37, 0, 11, 0,⟨−12,−18, 0, 0, 11; dimW 2; dimW 3; 2 3 5.
4. 1 − 5 2
3 − 3 2
; W: ⟨2,−5, 6,−3; W: ⟨5, 2, 0, 0,⟨−3, 0, 1, 0,⟨3, 0, 0, 2;
dimW 1; dimW 3; 1 3 4. 5. 1 − 1
3 5 3
2 3
2 ; W: ⟨3,−1, 5, 2, 6;
W: ⟨1, 3, 0, 0, 0,⟨−5, 0, 3, 0, 0,⟨−2, 0, 0, 3, 0,⟨−2, 0, 0, 0, 1; dimW 1; dimW 4; 1 4 5.
6.
1 0 26 17
0
0 1 11 34
0
0 0 0 1
; W: ⟨17, 0, 26, 0, ⟨0, 34, 11, 0, ⟨0, 0, 0, 1;
W: ⟨−52,−11, 34, 0; dimW 3; dimW 1; 3 1 4.
7.
1 0 26 17 − 1
17
0 1 11 34 − 11
17
0 0 0 0
; W: ⟨17, 0, 26,−1, ⟨0, 34, 11,−22;
W: ⟨−52,−11, 34, 0,⟨1, 11, 0, 17; dimW 2; dimW 2; 2 2 4.
8.
1 0 0 −2 3
0 1 0 3 −5
0 0 1 −4 7
; W: ⟨1, 0, 0,−2, 3, ⟨0, 1, 0, 3,−5, ⟨0, 0, 1,−4, 7;
W: ⟨2,−3, 4, 1, 0,⟨−3, 5,−7, 0, 1; dimW 3; dimW 2; 3 2 5.
Selected Answers to the Exercises 15
9.
1 0 0 0 17 16
0 1 0 0 − 1 8
0 0 1 0 3 16
0 0 0 1 − 5 8
; W: ⟨16, 0, 0, 0, 17, ⟨0, 8, 0, 0,−1, ⟨0, 0, 16, 0, 3, ⟨0, 0, 0, 8,−5;
W: ⟨−17, 2,−3, 10, 16; dimW 4; dimW 1; 4 1 5.
10.
1 0 0 −17 9
0 1 0 −29 16
0 0 1 −2 2
0 0 0 0 0
0 0 0 0 0
; W Span⟨1, 0, 0,−17, 9, ⟨0, 1, 0,−29, 16, ⟨0, 0, 1,−2, 2;
W Span⟨17, 29, 2, 1, 0, ⟨−9,−16,−2, 0, 1; dimW 3; dimW 2; 3 2 5.
11. a. Yes. b. Yes. c. No. d. Yes. e. No. f. No.
12. a. R
1 0 5 4
− 3 4 − 11
4
0 1 − 9 8 − 13
8 − 17
8
0 0 0 0 0
0 0 0 0 0
; b. ⟨−10, 9, 8, 0, 0,⟨6, 13, 0, 8, 0,⟨22, 17, 0, 0, 8
c. ⟨4, 0, 5,−3,−11,⟨0, 8,−9,−13,−17 d. dimW 2; dimW 3; 2 3 5. e. B1 is not a basis because the 2nd vector is parallel to the first. B2 is a basis because dimW 2 and these two vectors are not parallel to each other and both vectors are members of the Spanning set. B3 is a basis for W for the same reason.
13. a. R
1 − 2 3
0 0 − 11 18
0 0 1 0 − 1 2
0 0 0 1 5 6
0 0 0 0 0
e. R/
1 0 0 9 4
0 1 0 1 4
0 0 1 0
0 0 0 0
0 0 0 0
b. ⟨2, 3, 0, 0, 0,⟨11, 0, 9,−15, 18 c. ⟨18,−12, 0, 0,−11,⟨0, 0, 2, 0,−1,⟨0, 0, 0, 6, 5; d. dimW 3 and dimW 2; 3 2 5. f. B1 is a basis because the first 3 columns of R/ are linearly independent. B2 is not, because the 4th column is dependent on the first two. B3 is independent. Suppose v4 c1v1 c3v3 where neither c1 nor c3 is zero (notice, v4 is not parallel to either v1 or v3, so a dependence equation must involve both vectors). But v4 94 v1
1 4
v2. Setting these two equal, we would get a dependence equation for v1, v2 and v3, which is impossible. Similarly, B4 is independent.
16 Selected Answers to the Exercises
14. a. R
1 0 0 5 3
0 1 0 5 6
0 0 1 2 3
0 0 0 0
0 0 0 0
e. R/
1 0 7 0 9
0 1 −5 0 −3
0 0 0 1 2
0 0 0 0 0
b. ⟨−10,−5,−4, 6
c. ⟨3, 0, 0, 5,⟨0, 6, 0, 5,⟨0, 0, 3, 2 d. dimW 3 and dimW 1; 3 1 4. f. B1 is dependent, because the first three columns of R/ are dependent. B2 is independent, because c1, c2 and c4 of R/ are independent. B3 is independent. Suppose v4 c2v2 c3v3, again, where neither c2 nor c3 is zero. But we know that v3 7v1 − 5v2. Plugging this into the previous equation gives us a dependence equation for v4 with v1 and v2, which is impossible. B4 is independent but the reasoning is a bit more complicated. Suppose v5 c3v3 c4v4. Replace v3 with 7v1 − 5v2 as before, and distribute this over c3. Replace v5 with 9v1 − 3v2 2v4. Use the Uniqueness of Representation Property to get a contradiction.
15. a. ⟨1, 0, 5, 0, 4, ⟨0, 1,−4, 0, 3, ⟨0, 0, 0, 1,−6 b. ⟨4, 3, 8, 5,−5, ⟨5, 7,−3, 6, 5, ⟨3, 4,−1, 4, 0 c. ⟨−5, 4, 1, 0, 0, ⟨−4,−3, 0, 6, 1 d. B1 is not a basis; B2 is a basis; B3 is a basis.
16. a. ⟨3, 0, 0,−10,−2, ⟨0, 3, 0, 4,−1, ⟨0, 0, 3,−13,−2 b. ⟨5, 8,−3, 7,−4, ⟨3, 4,−2, 4,−2, ⟨−8,−9, 5,−7, 5 c. ⟨10,−4, 13, 3, 0,⟨2, 1, 2, 0, 3 d. B1 is not a basis; B2 is a basis; B3 is a basis.
17. a. ⟨44, 0, 0, 0, 81, ⟨0, 66, 0, 0, 25, ⟨0, 0, 44, 0,−29, ⟨0, 0, 0, 132,−487 b. ⟨7, 4,−3, 2, 9, ⟨3,−5, 2,−1, 6, ⟨6, 9,−7, 3, 8, ⟨4,−2, 5,−1, 7 c. ⟨−243,−50, 87, 487, 132 d. B1 is a basis; B2 is a basis; B3 is not a basis.
18. a. ⟨1, 0, 7,−4, 0, 2, ⟨0, 1,−5, 3, 0,−5, ⟨0, 0, 0, 0, 1, 8 b. ⟨3, 5,−4, 3, 3, 5, ⟨−2,−4, 6,−4,−2, 0, ⟨1,−1, 12,−7,−1,−1 c. ⟨−7, 5, 1, 0, 0, 0,⟨4,−3, 0, 1, 0, 0, ⟨−2, 5, 0, 0,−8, 1 d. B1 is a basis; B2 is a basis; B3 is not a basis; B4 is a basis.
The rref of
1 −1/2 2
−1/2 −1/2 1
1 1/2 0
is
1 0 0
0 1 0
0 0 1
, so the last 3 columns are linearly
independent. Thus, B4 is linearly independent. 19. a. No. b. Yes. c. Yes. d. No. e. No. 20. a. Yes. b. Yes. c. Yes. d. Yes. e. No. 21. a. Yes. b. Yes. c. Yes. d. No. e. Yes. 22. 22. a. Yes. b. Yes. c. No. d. Yes. e. Yes. 23. a. No. b. No. c. Yes. d. Yes. e. No. 24. a. No. b. Yes. c. No. d. Yes. e. No. 25. a. Yes. b. No. c. No. d. Yes. e. No. 26. a. Yes. b. Yes. c. Yes. d. Yes. e. No. 27. a. No. b. No. c. Yes. d. Yes. e. Yes. 36. a. True. b. True. c. False. d. True. e. False. f. True. g. True. h. True. i. False. j.
True. k. False. l. True. m. False. n. False. o. True. p. True. q. False. r. False. s. True. t. True. u. True.
Selected Answers to the Exercises 17
37. a. False. b. True. c. False. d. False. e. True. f. False. g. False. h. True. i. False. j. True. k. False. l. True. m. False. n. True. o. False. p. False. q. False. r. True. s. True. t. False.
18 Selected Answers to the Exercises
Chapter Two Exercises
2.1 Exercises
1. a. f is a function since every parent has a unique oldest child. b. g is not a function because x may not have any daughter at all. c. h is a function because every person has a unique mother. d. k is not a function because y may not have any brother at all. e. p is not a function because even though x has at least one child, none of the children of x may have any children of their own. f. q is a function because the father of y is unique, say call him z, and the mother of z is also unique.
2. a. ⟨−15, 38, 5. c. T 2 3
1 −5
4 1
.
3. a. ⟨−25,−6,−9. c. T 2 0 −5 0
0 3 1 −2
3 8 0 0
.
4. a. ⟨55,−21, 58, 84. c. T
3 2 −5
1 0 4
0 2 −7
4 9 0
5. a. ⟨23, 62,−10. c. T 5 −3 −2
4 −6 3
2 2 0
6. No. T is neither additive nor homogeneous. 7. No. T is neither additive nor homogeneous.
8. a. T
0 2
−5 4
3 −7
. b. T⟨x, y ⟨2y,−5x 4y, 3x − 7y c. ⟨−4,−43, 35.
9. a. T −3 2 0
5 7 4 . b. T⟨x, y, z ⟨−3x 2y, 5x 7y 4z c. ⟨−19, 35.
10. a. T
0 0 0 0 1
0 0 1 0 0
1 0 0 0 0
0 0 0 1 0
0 1 0 0 0
.
Selected Answers to the Exercises 19
b. T⟨x1, x2, x3, x4, x5 ⟨x5, x3, x1, x4, x2 c. ⟨9,−5, 3, 2, 0. 11. Tv1 ⟨6,−4, 17 and Tv2 ⟨−13, 10,−44.
.
x
y
1
1
T ( i )
T ( ij )T ( j )
2
x
y
Exercise 12
1
.
x
y 1
T ( i )
T ( ij ) T ( j )
Exercise 13 Exercise 14
x
y
1T ( i )
T ( j )
Exercise 15
T ( i )
T ( j )
T ( ij )
T ( ij )
.
.
. Exercise 16 Exercise 17
T ( i )
T ( ij )T ( j )
x
y
1
T ( i )
T ( ij )T ( j )
x
y
1
.
.
x
y
1
Exercise 18 Exercise 19
T ( i )
T ( j )
T ( ij )
T ( i )
T ( ij )
T ( j )
x
y 1
.
Exercise 20
T ( ij )
T ( j )
x
y
T ( i ) .
x
y
Exercise 21 Exercise 22 Exercise 23
T ( j )
x
y
T ( ij )
T ( j ) x
y
T ( i )
T ( i )
T ( ij ) T ( j )
T ( j )
T ( ij )
23. The box “collapsed” into a line, because the two columns are parallel.
x
y
v
uv
u x
y
Exercise 24
T ( u )
T ( v )
T ( uv ) 1
Exercise 25
1
x
y
v
uv
u
x
y
T ( u )
T ( v )
T ( uv )
1
.
.
1
26. a. Yes. b. No. c. No. d. Yes. e. No. f. No. g. Yes. h. No. i. No. j. No. k. No. l. Yes.
29. Sk
k 0 0
0 k 0
0 0 k
20 Selected Answers to the Exercises
2.2 Exercises
1. 3 /2 −1/2
1/2 3 /2 ; rot⟨5, 3
5 3 − 3 2
, 3 3 5
2
2. 4/5 −3/5
3/5 4/5 ; rot⟨5, 3 ⟨11/5, 27/5
3. −5/13 −12/13
12/13 −5/13 ; rot⟨5, 3 ⟨−61/13, 45/13
4. 12/13 −5/13
5/13 12/13 ; rot⟨5, 3 ⟨45/13, 61/13
5. − 1
2 2 − 2 − 1
2 2 2
1 2
2 2 − 1 2
2 − 2 ;
rot⟨5, 3 − 32 2 2 − 5 2 − 2 2 , 5
2 2 2 − 3
2 − 2 2
≈ ⟨−4. 685, 3. 471
6. −1/2 3 /2
− 3 /2 −1/2 ; rot⟨5, 3 −5 3 3 /2, −3 − 5 3 /2
7. 21/29 20/29
−20/29 21/29 ; rot⟨5, 3 ⟨165/29,−37/29
8. 3/5 4/5
−4/5 3/5 ; rot⟨5, 3 ⟨27/5,−11/5
9. −8/17 15/17
−15/17 −8/17 ; rot⟨5, 3 ⟨5/17,−99/17
10. − 41
841 840 841
− 840 841
− 41 841
; rot⟨5, 3 2315841 ,−
4323 841
≈ ⟨2. 75,−5. 14
11. projL 25/34 15/34
15/34 9/34 ; projL
9/34 −15/34
−15/34 25/34 ;
reflL 8/17 15/17
15/17 −8/17 ;
projL⟨3, 2 ⟨105/34, 63/34;
Selected Answers to the Exercises 21
projL⟨3, 2 ⟨−3/34, 5/34; reflL⟨3, 2 ⟨54/17, 29/17
12. projL 49/65 28/65
28/65 16/65 ; projL
16/65 −28/65
−28/65 49/65 ;
reflL 33/65 56/65
56/65 −33/65 ;
projL⟨3, 2 ⟨203/65, 116/65; projL⟨3, 2 ⟨−8/65, 14/65; reflL⟨3, 2 ⟨211/65, 102/65
13. projL 25/41 −20/41
−20/41 16/41 ; projL
16/41 20/41
20/41 25/41 ;
reflL 9/41 −40/41
−40/41 −9/41 ;
projL⟨3, 2 ⟨35/41,−28/41; projL⟨3, 2 ⟨88/41, 110/41; reflL⟨3, 2 ⟨−53/41,−138/41
14. projL 9/58 −21/58
−21/58 49/58 ; projL
49/58 21/58
21/58 9/58 ;
reflL −20/29 −21/29
−21/29 20/29 ;
projL⟨3, 2 ⟨−15/58, 35/58; projL⟨3, 2 ⟨189/58, 81/58; reflL⟨3, 2 ⟨−102/29,−23/29
15. projL 1 10
3 10
3 10
9 10
3
2 ; projL
9 10
− 3 10
− 3 10
1 10
; reflL − 4
5 3 5
3 5
4 5
;
projL⟨3, 2 ⟨9/10, 27/10; projL⟨3, 2 ⟨21/10,−7/10; reflL⟨3, 2 ⟨−6/5, 17/5
16. proj 129
13 −8 12
−8 25 6
12 6 20
; projL 129
16 8 −12
8 4 −6
−12 −6 9
;
refl 129
−3 −16 24
−16 21 12
24 12 11
17. proj 165
61 10 −12
10 40 30
−12 30 29
; projL 165
4 −10 12
−10 25 −30
12 −30 36
;
22 Selected Answers to the Exercises
refl 165
57 20 −24
20 15 60
−24 60 −7
18. proj 190
41 28 35
28 74 −20
35 −20 65
; projL 190
49 −28 −35
−28 16 20
−35 20 25
;
refl 145
−4 28 35
28 29 −20
35 −20 20
19. proj 134
25 0 −15
0 34 0
−15 0 9
; projL 134
9 0 15
0 0 0
15 0 25
;
refl 117
8 0 −15
0 17 0
−15 0 −8
20. proj 153
53 0 0
0 49 14
0 14 4
; projL 153
0 0 0
0 4 −14
0 −14 49
;
refl 153
53 0 0
0 45 28
0 28 −45
21. 0 −1
1 0 ; No, because of the −1.
22. reflL 119
−10 −15 6
−15 6 −10
6 −10 −15
−refl.
23. reflL 165
−57 −20 24
−20 −15 −60
24 −60 7
24. a. Tv ⟨2, 5 and Tw ⟨4,−3. c. it corresponds to reflL
Selected Answers to the Exercises 23
e.
1 0 0
0 0 1
0 1 0
is the matrix of the reflection across y z, and
0 0 1
0 1 0
1 0 0
is the
matrix of the reflection across x z. f. T⟨x1, x2, x3, x4 ⟨x1, x4, x3, x2; T exchanges the 2nd and 4th components of v.
27. 6x − 3y 8z 0. 28. a. 29 / 38 , 13 / 38 , 34 / 38 . The radicand in the numerator is the respective
diagonal entry.
b. 15 38
; −6 38
; 10 38
; c. cosi,j 15 377
; i,j cos−1 15 377
≈ 39. 420
cosi,k −6 986
; i,k cos−1 −6 986
≈ 101. 020; cosj,k 10 442
;
j,k cos−1 10 442
≈ 61. 600
2.3 Exercises
1. a. T1 T2⟨x, y, z ⟨5x − 2y 14z, 2x 3y − 4z. b. T1 T2 5 −2 14
2 3 −4
c. T1 3 −2 5
1 4 −7 and T2
2 0 9
1 −1 3
e. −4T1 −12 8 −20
−4 −16 28 −4T1.
2. a. T1 T2⟨x, y, z ⟨3x − 2y 4z, 2x − y − 4z, x 2y 3z,−3x − y z.
b.
3 −2 4
2 −1 −4
1 2 3
−3 −1 1
c. T1
1 −2 3
1 0 −4
0 2 0
1 −1 1
and T2
2 0 1
1 −1 0
1 0 3
−4 0 0
e.
3 −6 9
3 0 −12
0 6 0
3 −3 3
3. The matrices that do not exist are: b. A − B d. 7C 4A f. CB h. BE. The matrices that exist, and their sizes, are:
a. −2 −4 −3
6 7 −5 2 3
24 Selected Answers to the Exercises
c.
−3 −11
4 26
−29 −15
3 2
e.
−32 22
43 −19
−4 −7
3 2
g. 57 −40
−20 17 2 2
i.
3 31 −13
−2 −46 20
17 −27 17
3 3
j. 55 34
−19 15 2 2
k. 317 −118
−163 121 2 2
m.
−13 195 −91
26 −314 148
65 −367 183
3 3
o. 461 178
−167 −23 2 2
4. a.
1 8 −15
37 −52 −69
−28 −17 2
; 3 3
b.
56 5 −35 55
−1 −29 18 16
−3 −24 −13 39
−39 4 41 −63
; 4 4
Selected Answers to the Exercises 25
c.
5 −15 −15 13 70
93 −35 63 −49 88
−63 −15 −14 31 −4
; 3 5
d. does not exist
e.
13 −56 72
52 −31 −41
−63 50 10
37 −29 −60
; 4 3
f.
−50 −53 65 −25
23 1 0 10
64 26 −20 20
−11 −17 −12 26
−16 −6 20 −20
; 5 4
g.
41 −51 −84
−19 20 41
14 −17 −60
12 2 31
41 36 −9
; 5 3
h. does not exist
i.
89 59 −59 30
−17 −49 −21 0
−85 27 139 −58
71 6 −75 −4
; 4 4
j. does not exist.
k.
631 −225 362 −299 672
−237 −105 −101 163 194
14 −250 247 −54 272
−550 310 −477 312 −622
; 4 5
l. same as k
26 Selected Answers to the Exercises
m.
503 −1 −356
−139 −207 −326
−425 649 1340
306 83 −560
; 4 3 (same as part n)
o.
717 −153 −597
45 4173 2895
−713 626 1597
; 3 3
5. a. The codomain of T1 is 4, which is also the domain of T2. The domain of T2 ∘ T1 is 2 and the codomain is 3
b. This is not well defined. c. ⟨9x − 26y, 33x 9y,−6x 54y
d.
9 −26
33 9
−6 54
e. T2
3 0 0 −5
0 7 2 −1
0 0 6 9
; T1
3 −2
5 1
−1 3
0 4
;
T2T1
9 −26
33 9
−6 54
6. a. The codomain of one is the domain of the other, so both compositions are well-defined. T2 ∘ T1 : 3 → 3 and T1 ∘ T2 : 4 → 4. b. T2 ∘ T1⟨x, y, z ⟨9x 10y 7z, 16x − 8y 32z, 6x 9y − 12z, and T1 ∘ T2⟨x1, x2, x3, x4 ⟨9x1 35x2 4x3 − 29x4, 6x1 − 7x2 22x3 27x4, 3x1 6x3 4x
c. T2 ∘ T1 9 10 7
16 −8 32
6 9 −12
, T1 ∘ T2
9 35 4 −29
6 −7 22 27
3 0 6 4
0 7 −10 −19
d. T2
3 0 0 −5
0 7 2 −1
0 0 6 9
; T1
3 5 −1
2 −1 4
1 0 1
0 1 −2
;
T2T1
9 10 7
16 −8 32
6 9 −12
; T1T2
9 35 4 −29
6 −7 22 27
3 0 6 4
0 7 −10 −19
Selected Answers to the Exercises 27
7. a. The codomain of one is the domain of the other, so both compositions are well-defined. T2 ∘ T1 : 2 → 2 and T1 ∘ T2 : 5 → 5. b. T2 ∘ T1⟨x, y ⟨10x − 13y, 17x 26y, and T1 ∘ T2⟨x1, x2, x3, x4, x5 ⟨21x1 7x2 − 2x3 3x4 − 6x5, 21x2 − 20x3 16x4 − 25x5, 78x1 35x2 − 16x3 18x4 − 33x5, 54x1 12x3 − 6x4 6x5,−6x1 − 14x2 12x3 − 10x4 16x
c. T2 ∘ T1 10 −13
17 26 ; T1 ∘ T2
21 7 −2 3 −6
0 21 −20 16 −25
78 35 −16 18 −33
54 0 12 −6 6
−6 −14 12 −10 16
d. T2 3 7 −6 5 −8
9 0 2 −1 1
T1
1 2
3 −1
5 7
0 6
−2 0
3 7 −6 5 −8
9 0 2 −1 1 ; T2T1
10 −13
17 26 ;
T1T2
21 7 −2 3 −6
0 21 −20 16 −25
78 35 −16 18 −33
54 0 12 −6 6
−6 −14 12 −10 16
13. If A is m k, then B has to be k m. For both compositions to be defined, m must equal n.
2.4 Exercises
1. a.
11 −7 −1 11
−6 4 0 1
−7 13 8 4
b.
96 138
−54 −32
−72 5
c.
56 75
0 77
−40 −15
d.
40 63
−54 −109
−32 20
e.
96 138
−54 −32
−72 5
28 Selected Answers to the Exercises
f.
2 12
−3 −2
0 6
8 7
g.
48 59
−64 −132
5 8
h.
8 −4
−10 −23
37 −12
i.
48 59
−64 −132
5 8
j.
131 217
−16 −73
−21 −34
2. a. T1
2 −3 0
0 5 −7
1 −1 4
6 1 −1
; 4 3; T2 5 0 2 −1
2 8 −6 7 ; 2 4;
T3
1 2
1 −1
7 3
4 1
1 5
; 5 2
b. T2 ∘ T1⟨x, y, z ⟨6x − 18y 9z, 40x 47y − 87z.
c. 6 −18 9
40 47 −87 ; 2 3; d. same as c.
e. T3 ∘ T2
9 16 −10 13
3 −8 8 −8
41 24 −4 14
22 8 2 3
15 40 −28 34
5 4 ;
f. T3 ∘ T2 ∘ T1
86 76 −165
−34 −65 96
162 15 −198
64 −25 −51
206 217 −426
5 3.
3. a. T1 8/17 15/17
15/17 −8/17 ; T2
3/5 4/5
−4/5 3/5 ;
Selected Answers to the Exercises 29
T3 9/58 −21/58
−21/58 49/58 .
b. T2 ∘ T1 84 85
13 85
13 85 − 84
85
; T1 ∘ T3 − 243
986 567 986
303 986
− 707 986
c. T3 ∘ T2 ∘ T1 483 4930
1881 4930
− 1127 4930
− 4389 4930
; T1 ∘ T3 ∘ T2 − 2997
4930 729
4930
3737 4930
− 909 4930
;
we get different answers.
4. T1 2 −3 1
4 −5 −7 2 3, T2
5 −4
1 −3
7 2
3 2,
T1 ∘ T2 14 3
−34 −15 2 2, T2 ∘ T1
−6 5 33
−10 12 22
22 −31 −7
3 3.
5. A2 44 −35
−25 39 , A3
−307 378
270 −253 , A4
2811 −2905
−2075 2396 .
pA 4I2 − 6A 5A2 − 2A3 7A4 20, 533 −21, 308
−15, 220 17, 489 . Reminder: the first
term is 4I2.
6. A2
3 −8 −16
0 1 −6
4 24 51
, A3
−5 −56 −118
9 −25 −42
25 180 349
, pA
−15 −72 −170
39 −59 −54
23 268 495
7. a. We have two non-zero, non-parallel vectors. b. ⟨217, 579,−694 8. a. The rref of the matrix with the 3 vectors as columns is I3. b. ⟨18, 192 ,
57 2
,−7, 59 2
16. a. −1 0
0 1
cos −sin sin cos
; rotate 2 by , then reflect 2 across the y-axis.
b. cos −sin sin cos
1 0
0 −1 ; reflect 2 across the x-axis, then rotate 2 by .
19. Rotating 2 by , followed by another rotation by results in a net rotation by . Similarly, rotating 2 by , followed by another rotation by results in a net rotation by , which is the same as .
30 Selected Answers to the Exercises
2.5 Exercises
1. a. T1
3 1 −7 8
2 2 −2 −4
−2 1 8 −17
b. R1
1 0 −3 5
0 1 2 −7
0 0 0 0
c. ⟨3,−2, 1, 0,⟨−5, 7, 0, 1 d. nullityT1 2 e. T1 is not 1-1. f. ⟨3, 2,−2,⟨1, 2, 1 g. rankT1 2 h. T1 is not onto. i. 2 2 4.
2. a. T2
3 −6 5
2 −4 7
−5 10 3
−1 2 8
b. R2
1 −2 0
0 0 1
0 0 0
0 0 0
c. ⟨2, 1, 0
d. nullityT2 1; e. T2 is not 1-1. f. ⟨3, 2,−5,−1,⟨5, 7, 3, 8 g. rankT2 2. h. T2 is not onto. i. 2 1 3.
3. a. T3
−5 −7 2
−2 1 16
3 −2 −26
b. R3
1 0 −6
0 1 4
0 0 0
c. ⟨6,−4, 1 d. nullityT3 1; e. T3 is not 1-1. f. ⟨−5,−2, 3,⟨−7, 1,−2 g. rankT 2. h. T3 is not onto. i. 2 1 3 j. The kernel is a line with direction ⟨6,−4, 1, and the range is a plane with equation x − 31y − 19z 0 k. The kernel is not necessarily orthogonal to the range (columnspace). The kernel is always orthogonal to the rowspace.
4. a. ⟨−5, 2, 1 b. 1 c. T is not one-to-one. d. ⟨2, 3, 3,−3, 3,⟨3, 4, 5, 2, 10 e. 2. f. T is not onto. g. not full-rank. h. 2 1 3.
5. a. there is no basis for the kernel of T. b. 0 c. T is one-to-one. d. ⟨2, 3, 3,−3, 3,⟨3, 4, 5, 2, 10,⟨4, 7, 5,−18,−5 e. 3 f. T is not onto. g. full-rank. h. 3 0 3.
6. a. ⟨−4,−9, 1, 0, 0,⟨5, 3, 0, 1, 0,⟨−2, 1, 0, 0, 1 b. 3 c. T is not one-to-one. d. ⟨3,−5,−8,⟨−2, 3, 5 e. 2 f. T is not onto. g. not full-rank. h. 2 3 5.
7. a. ⟨−4,−9, 1, 0, 0,⟨5, 3, 0, 1, 0 b. 2 c. T is not one-to-one. d. ⟨3,−5,−8,⟨−2, 3, 5,⟨8,−13,−20 e. 3 f. T is onto. g. full-rank. h. 3 2 5.
8. a. ⟨−2,−3, 1, 1, 0,⟨1,−2, 5, 0, 1 b. 2 c. T is not one-to-one. d. ⟨3,−5,−8,⟨−2, 3, 5,⟨−2, 9, 4 e. 3 f. T is onto. g. full-rank. h. 3 2 5.
9. a. ⟨−4,−9, 1, 0, 0,⟨−3, 8, 0,−5, 1 b. 2 c. T is not one-to-one. d. ⟨3,−5,−8, 6,⟨−2, 3, 5,−3,⟨−2, 9, 10,−8 e. 3 f. T is not onto. g. not full-rank. h. 3 2 5.
10. a. ⟨3, 1, 0, 0, 0,⟨7, 0,−5, 1, 0 b. 2 c. T is not one-to-one. d. ⟨3,−5,−2, 2,⟨6,−7,−3, 5,⟨−2, 9, 7,−8 e. 3 f. T is not onto. g. not full-rank. h. 3 2 5.
11. a. ⟨−2, 1,−3,−5, 1 b. 1 c. T is not one-to-one. d. ⟨3,−5,−2, 2,⟨6,−7,−3, 5,⟨−2, 3, 4, 7,⟨−1,−4, 3,−2 e. 4 f. T is onto. g. full-rank. h. 4 1 5.
12. a. ⟨−5, 2, 1, 0, 0 b. 1 c. T is not one-to-one.
Selected Answers to the Exercises 31
d. ⟨3,−5,−2, 2,⟨6,−7,−3, 5,⟨−2, 3, 4, 7,⟨−1,−4, 3,−2 e. 4 f. T is onto. g. full-rank. h. 4 1 5.
13. a. ⟨−72, 25, 45, 0,⟨−36, 35, 0, 45 b. 2 c. T is not one-to-one. d. ⟨15, 30,−10,−5,−15,⟨72, 63, 27,−54, 0 e. 2 f. T is not onto. g. not full-rank. h. 2 2 4.
14. a. there is no basis for the kernel of T b. 0 c. T is one-to-one. d. ⟨1, 3,−1,−5, 5,⟨2, 6, 7,−4, 0,⟨−6, 3,−3, 2,−4,⟨−4,−5,−2, 3, 1 e. 4 f. T is not onto. g. full-rank. h. 4 0 4.
15. a. ⟨−4, 3,−2, 1 b. 1 c. T is not one-to-one. d. ⟨5, 2,−6,−2, 1,⟨7,−1,−3, 3, 0,⟨2, 3,−5, 1,−1 e. 3 f. T is not onto. g. not full-rank. h. 3 1 4.
16. a. ⟨3, 1, 0, 0,⟨−4, 0, 2, 1 b. 2 c. T is not one-to-one. d. ⟨2, 3, 2, 5,⟨3, 1, 5, 4 e. 2 f. T is not onto. g. not full-rank. h. 2 2 4.
17. a. ⟨5,−3,−8, 1 b. 1 c. T is not one-to-one. d. ⟨4, 5,−6, 5,⟨2, 9,−7, 6,⟨1,−2,−1, 3 e. 3 f. T is not onto. g. not full-rank. h. 3 1 4.
18. a. ⟨5, 1, 0, 0, 0,⟨−9, 0, 7, 1, 0 b. 2 c. T is not one-to-one. d. ⟨−3, 2, 5, 0,−4,⟨−5,−1, 2,−3,−7,⟨12,−4, 0,−25, 37 e. 3 f. T is not onto. g. not full-rank. h. 3 2 5.
19. a. ⟨7,−5, 1, 0, 0 b. 1 c. T is not one-to-one. d. ⟨−3, 2, 4, 0,−3,⟨−5,−1, 6,−1,−4,⟨2,−4,−5,−5, 3,⟨−5,−1, 2,−3,−7 e. 4 f. T is not onto. g. not full-rank. h. 4 1 5.
20. a. ⟨−7, 2,−3, 1, 0,⟨5,−3, 2, 0, 1 b. 2 c. T is not one-to-one. d. ⟨−3, 2, 4, 0,−3,⟨−5,−1, 2,−3,−7,⟨2,−4,−5,−5, 3 e. 3 f. T is not onto. g. not full-rank. h. 3 2 5.
22. a. b. L c. L d. e. 03 f. 3
31. The three image vectors are linearly dependent: 8 5 ⟨2,−3, 4,−1, 7 − 3
5 ⟨−3, 2,−1, 4, 2 ⟨5,−6, 7,−4, 10, so
8 5 ⟨1,−2, 1 − 3
5 ⟨0,−1, 3 − ⟨0,−2, 5 8
5 ,− 3
5 ,− 26
5 is a non-zero vector in kerT.
32. a. True. b. False. c. True. d. False. e. False. f. True. g. True. h. True. i. False. j. True. k. False. l. False. m. True. n. True.
2.6 Exercises
1. − 1
2 0
0 1 3
2.
1 5
7 20
0 − 1 4
3. 0 1
6 1 4
0
4. 4 −9
−3 7 5.
−1 −2
− 4 3 − 7
3 6.
3 2
0
0 − 3 8
32 Selected Answers to the Exercises
7.
3 5
4 5
− 1 2 − 1
2
8. not invertible. 9.
11 19
− 5 57
14 19
4 57
10. − 27
124 11 124
12 31
2 31
11.
105 179
− 24 179
10 179
100 179
12. 1 24
− 6 30
2 15 −2 3 13.
cos sin
−sin cos which is the matrix of the clockwise
rotation by .
14. cos sin
sin −cos 15. 1
5 3e−3x e2x
−2e−4x ex 16. 1
2 60x 10x 4x
15x −6x
17. coshx −sinhx −sinhx coshx
18. not invertible. 19. a2 − b2 2ab
2ab b2 − a2
20. The projection operator (Exercise 18) is not invertible because the kernel consists of more than just the zero vector. The reflection operator (Exercise 19) is invertible because the kernel is only the zero vector. Furthermore, notice that the inverse is itself, for the reason that the reflection of the reflection of a vector is the original vector.
21. T 3 −7
−4 9 ; T−1
−9 −7
−4 −3 ; T−1⟨x, y ⟨ − 9x − 7y, − 4x − 3y
22. T is not invertible. 23. T 3 5
5 9 ; T−1
9 2
− 5 2
− 5 2
3 2
;
T−1⟨x, y ⟨9x/2 − 5y/2, − 5x/2 3y/2.
24. T 2 3
5 3
4 3 − 1
3
;T−1 3 22
15 22
6 11
− 3 11
;
T−1⟨x, y ⟨3x/22 15y/22, 6x/11 − 3y/11.
26. a. 6 −21
10 −35 ; No. e.
31 −27
−59 69 ; Yes. f.
31 124
−93 −372 ; No. g.
No. b. You will never get an invertible matrix. 27. b. not invertible. It is not one-to-one.
Selected Answers to the Exercises 33
2.7 Exercises
Note: answers vary for (b) and (c) in Exercises 1 to 12, so only answers to (a) are provided.
1. 2 − 7
3
−1 4 3
2. 7 11 − 3
11
10 11
2 11
3.
− 3 5
4 5
1
1 −1 −1 9 5 − 12
5 −2
4. not invertible. 5.
1 3
1 2 − 11
6
0 − 1 4
1
0 0 1 2
6.
2 0 0 3 4
3 2
0 13 12
1 6 − 1
3
7.
5 31
− 2 31 − 4
31
− 1 62
19 62
7 62
8 31
3 31
6 31
8.
8 27
2 27
11 27
7 27
− 5 27
13 27
− 4 9 − 1
9 − 1
9
9.
3 7
1 7
4 7
1 −1 2
− 9 7 − 3
7 2 7
10.
−1 3 2
11 25 6
0 1 2
2 1 6
0 0 1 2 3
0 0 0 1 3
11. not invertible.
12.
4 7
11 14
− 13 14
− 6 7
1 7
4 7
− 6 7 − 5
7
5 7
6 7
− 9 7 − 11
7
− 2 7 − 9
14 3 14
3 7
14.
55 31
− 73 62
26 31
15.
− 20 9
− 31 9
− 2 3
16.
4 7 − 24
7
−4 −18 16 7 − 26
7
34 Selected Answers to the Exercises
17.
− 253 6
− 37 6
− 11 3
− 1 3
18.
− 62 7
179 14
− 61 7
60 7
− 102 7
132 7
24 7
− 121 14
19. a. Multiply row 2 of A by −5. b. Multiply row 3 of A by −2/5. c. Add 3 times row A to row 2 of A. d. Add 7 times row 2 of A to row 3 of A. e. Exchange rows 1 and 3 of A. f. Subtract 4 times row 3 of A from row 1 of A.
20. a. Subtract 3 times row 4 of A from row 2 of A. b. Exchange rows 2 and 4 of A. c. Multiply row 3 of A by 3/2. d. Multiply row 4 of A by 9. e. Add 5 times row 2 of A to row 4 of A. f. Exchange rows 1 and 4 of A.
27. a. Subtract 3 times column 2 of A from column 4 of A. b. Exchange columns 2 and 4 of A. c. Multiply column 3 of A by 3/2. d. Multiply column 4 of A by 9. e. Add 5 times column 1 of A to column 3 of A. f. Exchange columns 1 and 4 of A.
2.8 Exercises
1. a. A−1 −3 5
2
2 − 3 2
; B−1 −1 4
3
−2 7 3
b.
17 3
− 9 2
32 3
− 17 2
c. 51 −27
64 −34 d.
17 3
− 9 2
32 3
− 17 2
7. A−1 BX−1 and B−1 X−1A. 9. B−1 is obtained from A−1 by exchanging columns 1 and 3 of A−1, followed by exchanging columns 2 and 5.
10. a. B
5 −2 1 0 0
−4 0 7 0 0
3 −9 −8 0 0
0 0 0 3 −7
0 0 0 −2 4
; C
3 −7 0 0
−2 4 0 0
0 0 3 −7
0 0 −2 4
b. The entries don’t match because the matrices are in opposite locations.
Selected Answers to the Exercises 35
c. A2 ⊕ A3
5 −2 1 0 0
−4 0 7 0 0
3 −9 −8 0 0
0 0 0 −4 5
0 0 0 7 −3
;
A1 ⊕ A2 ⊕ A3 A1 ⊕ A2 ⊕ A3
3 −7 0 0 0 0 0
−2 4 0 0 0 0 0
0 0 5 −2 1 0 0
0 0 −4 0 7 0 0
0 0 3 −9 −8 0 0
0 0 0 0 0 −4 5
0 0 0 0 0 7 −3
;
d.
8 −2 −1 0 0 0
4 6 −7 0 0 0
−3 5 9 0 0 0
0 0 0 5 0 0
0 0 0 0 0 9
0 0 0 0 −2 −5
; 6 6 e. Only B, with blocks B1 3 −7
−2 4
and B2 8 −1
0 5 .
11. a. A 7 12
−3 −5 ; A−1
−5 −12
3 7 . b. A
a b
y x , and
A−1 x −b
−y a
both have only integer entries.
c. 5 8
3 5 , with inverse
5 −8
−3 5 d.
3 −7
−7 16 , with inverse
−16 −7
−7 −3 .
Other answers are possible by switching entries.
36 Selected Answers to the Exercises
2.9 Exercises
1. a. lower triangular. b. all of the above. c. symmetric. d. all of the above. e. all of the above.
2. a.
2 3 4 5
3 4 5 6
4 5 6 7
5 6 7 8
b. Symmetric 3. a.
−12 −21 9 −6 0
18 −4 2 8 12
−35 −21 −14 63 7
b.
−27 −10 −14
−3 8 21
9. a.
24 27 −43
0 −6 8
0 0 28
12. a.
Te1 3e1,
Te2 −5e1 2e2, and
Te3 4e1 e2 − 7e3.
c.
1/3 5/6 13/42
0 1/2 1/14
0 0 −1/7
b. v1 13 e1, v2 5 6
e1 12 e2, v3 13 42
e1 114 e2 − 1 7
e3.
Selected Answers to the Exercises 37
Chapter Three Exercises
3.1 Exercises 5. There are no negatives for the vectors, even though there is a zero vector. 6. Not closed under scalar multiplication. 7. Not closed under addition: for example, identity plus its negative yields zero matrix,
which is not invertible. 8. −3 ⊙ ⟨5,−2 ⟨−15,−2. All Axioms are valid except for Axiom 7, so this is not a vector
space. 9. −3 ⊙ ⟨5,−2 ⟨15,−6. All Axioms are valid except for Axioms 9 and 10, so this is not a
vector space. 10. ⟨7,−3 ⊕ ⟨2, 6 ⟨7, 4. All Axioms are valid except for Axioms 7 and 8, so this is not a
vector space. 11. ⟨7,−3 ⊕ ⟨2, 6 ⟨9,−9. Invalid axioms: 3, 4, 5, 6, 7 and 8; not a vector space. 12. ⟨7,−3 ⊕ ⟨2, 6 ⟨−9,−3. Invalid axioms: 4, 5, 6 and 7; not a vector space. 13. ⟨7,−3 ⊕ ⟨2, 6 ⟨13,−1. Invalid axioms: 3, 4, 5, 6, and 7; not a vector space. 14. ⟨7,−3 ⊕ ⟨2, 6 ⟨9, 6 and −3 ⊙ ⟨5,−2 ⟨−15, 12. Invalid axioms: 4, 5, 6, 7, 9 and
10; not a vector space. 15. ⟨7,−3 ⊕ ⟨2, 6 ⟨9, 6 and −3 ⊙ ⟨5,−2 ⟨−30, 6. Invalid axioms: 4, 5, 6, 7, 9 and
10; not a vector space. 16. ⟨7,−3 ⊕ ⟨2, 6 ⟨3, 9 and −3 ⊙ ⟨5,−2 ⟨6,−15. Invalid axioms: 4, 5, 6, 7, 9 and
10; not a vector space. 17. ⟨7,−3 ⊕ ⟨2, 6 ⟨−9,−3 and −3 ⊙ ⟨5,−2 ⟨−15,−6. Invalid axioms: 4, 5, 6, 7, 9
and 10; not a vector space. 18. ⟨7,−3 ⊕ ⟨2, 6 ⟨9, 0 and −3 ⊙ ⟨5,−2 ⟨−15, 0. Invalid axioms: 5, 6, and 10; not a
vector space. 19. ⟨7,−3 ⊕ ⟨2, 6 ⟨7, 6 and −3 ⊙ ⟨5,−2 ⟨−13, 3. Invalid axioms: 8, 9, and 10; not a
vector space. However, there is a zero vector and negatives: 0V ⟨2,−3 and −⟨x1, y1 ⟨4 − x1,−6 − y1.
20. This is a vector space!
3.2 Exercises 1. Yes, a member. −7 19x − 47x2 36 3x − 4x2 − 55 − 2x 7x2 2. Yes, a member. 105 − 28x 39x2 9x3 72 − 4x 5x3 137 3x2 − 2x3 3. Yes, a member. 2x
2 − 7x − 10 x3
2x − 7 x2 − 10
x3 4. Not a member. 5. Yes, a member.
4x 25 x 1x − 2
11 x − 2
− 7 x 1
. 6. Not a member. 7.
x x2 − 1
1 2x − 1
1 2x 1
10. dependent 11. independent 12. dependent 13. independent 14. dependent 15. dependent 16. dependent
17. dependent 18. independent 19. independent 20. independent 21. independent 22. dependent 23. dependent
38 Selected Answers to the Exercises
24. dependent 25. independent 26. dependent 27. independent 28. dependent 29. dependent 30. dependent
31. dependent 32. dependent 33. dependent 34. independent. 39. d. independent
3.3 Exercises
4. a. Ex x2n | n ∈ ℕ 5. Ox x2n1 | n ∈ ℕ 6. a. S 1 x n1
| n ∈ ℕ 7. a.
0, d. fx 1x 1 is a legitimate (constant) function, and we do not care if the functions in S are one-to-one or not.
8. a. S x 1
n2 | n ∈ ℕ
9. independent 10. dependent (the logarithm requires a positive base b ≠ 1. 11. independent
12. dependent; S ⊂ ℙn, so once you have n 1 of these functions, they are definitely dependent; on the other hand, the set S in Exercise 11 is not contained in a single ℙn because there is a polynomial of any degree n in that S.
13. independent; take a limit at a vertical asymptote to show that the coefficient for that term must be 0.
14. independent 15. independent 16. independent 17. independent 18. independent 19. dependent (check out first six vectors) 20. independent
27. a. 1/6, −1/6, 7/6, −7/6, 11/6, −11/6, 13/6, 1/7, −1/7, 2/7, −2/7, 3/7, −3/7, 4/7, 1/8, −1/8, 3/8, −3/8, 5/8, −5/8, 7/8 b. 1/4, −1/3, 3/2, 2, −2, −3/2, 2/3, −1/4, 1/5, 1/6, −1/5, 3/4, −2/3, 5/2, 3, −3, −5/2, 4/3, −3/4, 2/5, −1/6, −1/7,
1/8, −1/7, 5/6. c. k i j − 1. 28. a. fx b − ax a f. fx x − a h. fx −x b k. fx −x 1 l.
fx −x − a b −x a b
m. fx
0 if x 0
−x 3 2
if x ∈ 1 2
, 1
−x 3 4
if x ∈ 1 4
, 1 2
−x 3 8
if x ∈ 1 8
, 1 4
−x 3 2n1
if x ∈ 1 2n1
, 1 2n
n.
10.750.50.250
1
0.75
0.5
0.25
0
x
y
x
y
Note: the top of each line segment should be an open hole, and the bottom should be a solid dot, and the graph keeps following the pattern as we get closer to the origin, where f0 0.
Selected Answers to the Exercises 39
3.4 Exercises 2. Yes, because every diagonal matrix is also symmetric. 3. A possible basis is ⟨1, 1, . . . , 1. The subspace is 1-dimensional. 4. Yes, it is a 1-dimensional subspace, with possible basis ⟨1, 2, . . . , n 5. No. It is not closed under either addition or scalar multiplication, although it does contain
the zero vector. 6. A possible basis for this 2-dimensional subspace is 1,−24x − 9x2 5x3. 7. A possible basis for this 2-dimensional subspace is −5 − 7x 8x2, 19 − 17x 2x3. 8. A possible basis for this 2-dimensional subspace is 1 2x,−4 x2. 9. A possible basis for this 2-dimensional subspace is −1 2x,−1 x2. 10. A possible basis for this 3-dimensional subspace is −2 x, 13 − 3x2,−10 x3. 11. It does not contain the zero vector. 12. A possible basis for this 1-dimensional subspace is 22 − 10x x2 x3. 13. A possible basis for this 1-dimensional subspace is 2e2x − 3e3x e5x. 14. A possible basis for this 2-dimensional subspace is −2e2x e3x,−4e2x e5x. W1 is a
subspace of W2. 15. A possible basis for this 2-dimensional subspace is:
2 − 1 sinx cosx,− 2 sinx tanx 16. c. It doesn’t contain the zero function zx. 17. Another hint: think of the factors of such a member of W. The subspace is 1-dimensional. 18. The sum of px x 2 and qx x − 3, which are both in W, is rx 2x − 1, which
is not in W. Can you come up with a counterexample where both p and q are quadratics? 19. Yes. 20. Yes. 21. Yes. 22. Yes. 23. No. The set is dependent, even though S is a
subset of W. 24. Yes. 25. No. This polynomial is not in W. 26. Yes. 27. Yes. 32. 2-dim 36.
dimDiagn n. 38. dimUppern nn 1/2. 39. The transpose of the basis vectors you found in
Exercise 38 will form a basis for Lowern, so the two spaces have exactly the same dimension.
41. The basis should have two kinds of matrices: those which are all 0 except for a single 1 on the main diagonal (thus there are n of these), and those which are all 0 except for a single 1 in row i, column j, as well as in row j, column i, where i ≠ j. There are 1 2 n − 1 of these. Thus there are 1 2 n − 1 n nn 1/2 members of this basis, which is dimSymn.
42. b. Possible answer: 1 0
0 1 ,
0 1
1 0 ; dimBisym2 2
d. Possible answer:
1 0 0
0 0 0
0 0 1
,
0 1 0
1 0 1
0 1 0
,
0 0 1
0 0 0
1 0 0
,
0 0 0
0 1 0
0 0 0
;
dimBisym3 4.
40 Selected Answers to the Exercises
e.
a b c d
b e f c
c f e b
d c b a
; dimBisym4 6 f.
a b c d e
b f g h d
c g i g c
d h g f b
e d c b a
;
dimBisym5 9 Use one matrix for every distinct letter.
43. d. Possible answer: I2, 1 5
−7 0 ; it is 2-dimensional.
45. e. Use the Ordinary Comparison Test. f. D does not contain the zero vector (zero series), which is absolutely convergent. g. It is not closed under vector addition.
3.5 Exercises 1. a. 21. b. 3 c. 1 2. a. ⟨0, 3/5, 1/2 b. ⟨1, 7/25, 1/2 c. 0, 3/4, 1/ 3 3. a. ⟨−66, 6 b. x 3x − 1 or x2 2x − 3. 4. a. ⟨−996, 156,−84 b.
x 5x − 3x 2 5. a. ⟨117, 13, 18 b. zx 6. a. ⟨6, 28,−26 7. a. ⟨−33,−2,−10, 16/3 8. a.
12x 10 9. a. 3x4 2x3 − 7x2 10. a. x3 x2 − 7x 11. a. −5e−x − 6e2x d. zx e.
rangeD W. 12. a. 7ex sinx ex cosx d. zx e. rangeD W. 13. a. 3e−3x sin2x 37e−3x cos2x d. zx e. rangeD W. 14. a. 33e5x − 10xe5x d. zx e. rangeD W. 15. a. 20x2e−4x − 18xe−4x 30e−4x d. zx e. rangeD W. 16. a. −4 ln 5x2 5x 9ln 5 − 8x 5x 9 − 2ln 55x d. zx e. rangeD W. 17. a. 6x2 − 16x 3 d. 1 e. 1, x, x2. 18. a. −18x sin2x 8x cos2x − 12 sin2x − cos2x d. zx e. rangeD W. 19. a. 27 sinx − cosx 20. a. 120e4x sin3x 102e4x cos3x 21. a. ac1 − bc2eax sin bx ac2 bc1eax cos bx 22. a. −4c1e−4x 3c2e3x 5c3e5x d. 91c1e−4x 64c3e5x e. e3x f. e−4x, e5x 26. a.
4 0
−3 1
5 −7
3.6 Exercises 1. a. ⟨−13/2, 19/2, 8 c. ⟨−1/2, 1/2, 1. 2. b. ⟨3/2, 27/2, 83,−545/3 3. a. −1/ 2 , 1/ 2 b. ⟨4/5, 3/5 c. ⟨−12/13, 5/13 d. ⟨20/29, 21/29
Selected Answers to the Exercises 41
4. a. 1 −3 9
1 1 1 c. ⟨82, 6 5. a.
1 −5 25 −125
1 3 9 27
1 −2 4 −8
c. ⟨−1285, 179,−91
6. a.
1 −5 25
1 3 9
1 −2 4
c. ⟨91, 27, 22 7. a.
1 1 1
0 1 −4
2 1 8
c. ⟨6,−33, 59
8. a.
1 −2 4
0 1 2
0 0 2
1 1/2 1/3
c. ⟨42, 9, 14, 23/6 9. a. 0 0 2 0
0 0 0 6 c. 42x − 16
10. a.
0 0 0
1 0 0
0 1/2 0
0 0 1/3
c. 7 3
x2 − 5 2
x2 4x 11. a. −1 0
0 2 b. −5e−x − 6e2x
12. a. 1 −1
1 1 b. 7ex sinx ex cosx 13. a.
−3 −2
2 −3 b. 3e−3x sin2x 37e−3x cos2x
14. a. 5 0
1 5 b. − 10 xe5x 33 e5x
15. a.
−4 0 0
2 −4 0
0 1 −4
b. 20x2e−4x − 18xe−4x 30e−4x
16.
ln5 0 0
2 ln5 0
0 1 ln5
b. − 4 ln5x2 5x 9 ln5 − 8x 5x −2 ln5 95x
17.
0 1 0 0
0 0 2 0
0 0 0 3
0 0 0 0
b. 6x2 − 16x 3 18.
0 −2 0 0
2 0 0 0
1 0 0 −2
0 1 2 0
b. −18x sin2x 8x cos2x − 12 sin2x − cos2x
19. b. 0 −m
m 0 20. b. DB Diagk1, k2, . . . , kn c. a diagonal matrix
42 Selected Answers to the Exercises
21. a −b
b a 22.
k 0 0
2 k 0
0 1 k
c. kxnekx nxn−1ekx
23. a. 1 3
−3 1 b. 27 sinx − cosx c. 13
5 sinx − 9
5 cosx
24. a. −3 −15
15 −3 c. −96e4x sin3x − 66e4x cos3x
25. a. 45x2 6x − 20 d. 2 −1 4 −2
0 0 4 6
0 0 0 9
26. a. −11x3 − 36x2 60x − 41 d.
−5 3 0
2 −5 6
0 3 −5
−1 2 0 27. a. ⟨95,−15,−6. b. 365x − 211. c. T1 4x − 2, Tx 21x − 7, and
Tx2 66x − 36.
d. TS,S/ −2 −7 −36
4 21 66
28. a. ⟨−11, 3 b. − 46x2 63x 126 c. T1 5x2 − 6x − 9; Tx −7x2 11x 27 d. −9 27
−6 11
5 −7 29. a. ⟨69/2,−14,−3. b. 311
2 − 167x 59
2 x2. c. T1 9
2 − 3x 1
2 x2,
Tx 25 2 − 10x 3
2 x2, and Tx2 83
2 − 46x 17
2 x2. d.
9/2 25/2 83/2
−3 −10 −46
1/2 3/2 17/2
31. proj 1122
113 −21 24
−21 73 56
24 56 58
; refl 161
52 −21 24
−21 12 56
24 56 −3
;
projL 1122
9 21 −24
21 49 −56
−24 −56 64
Selected Answers to the Exercises 43
32. proj 183
58 15 −35
15 74 21
−35 21 34
; refl 183
33 30 −70
30 65 42
−70 42 −15
;
projL 183
25 −15 35
−15 9 −21
35 −21 49
33. proj 130
26 2 −10
2 29 5
−10 5 5
; refl 115
11 2 −10
2 14 5
−10 5 −10
;
projL 130
4 −2 10
−2 1 −5
10 −5 25 34. a. choose ⟨2, 0, 3 and ⟨0, 1, 0 (note that the 2nd vector satisfies the equation);
f. proj 113
4 0 6
0 13 0
6 0 9
h. refl 113
−5 0 12
0 13 0
12 0 5
i. projL 113
9 0 −6
0 0 0
−6 0 4
35. d. C
−c 0 a
0 1 0
a 0 c
is one possible answer. 37. S / w 1, w 2, w 4;
w 3 4w 1 − 3w 2 38. S / w 1, w 2, w 4; w 3 −4w 1 3w 2; w 5 2w 1 − 5w 2 7w 4 39. S / w 1, w 2, w 5; w 3 4w 1 9w 2; w 4 5w 1 8w 2; w 6 −3w 1 4w 2 − 7w 5 40. S / w 1, w 2, w 4; w 3 4w 1 − 3w 2; w 5 6w 1 − 3w 2 − 4w 4
42. c. SuB,B/ 0 −a/c
1 −b/c d.
0 −3/5
1 2/5
44 Selected Answers to the Exercises
3.7 Exercises 1. a. No. b. Yes, because dimℙ2 dim4.
c.
1 0 0
0 1 0
0 0 1
0 0 0
d. kerT zx, so it has no basis, and nullityT 0. e. rangeT has basis ⟨1, 0, 0, 1,⟨−2, 1, 0, 1/2,⟨4, 2, 2, 1/3, and rankT 3 f. T is one-to-one but not onto. g. 3 0 3 dimℙ2 h. px 4 − 7x 5x2 is the only such polynomial.
2. a. Yes, because dimℙ3 dimℙ1. b. No.
c. 0 0 1 0
0 0 0 1
d. kerT has basis 1, x and nullityT 2. e. rangeT has basis x2, x3 and rankT 2. f. T is neither one-to-one nor onto. g. 2 2 4 dimℙ3
3. a. No. b. Yes, because dimℙ2 dimℙ3.
c.
1 0 0
0 1 0
0 0 1
0 0 0
d. kerT zx, so it has no basis and nullityT 0. e. rangeT has basis x, x2, x3 (we can clear the fractions) and rankT 3. f. T is one-to-one but not onto. g. 0 3 3 dimℙ2.
4. a. Yes, because dimℙ3 dimℙ2. b. No.
c.
1 − 1 2
0 0
0 0 1 0
0 0 0 1
d. kerT has basis 1 2x and nullityT 1. e. rangeT has basis 2, 4 4x,−2 6x 9x2 or 1, x, x2; either basis is acceptable because rankT 3. f. T is not one-to-one but T is onto. g. 3 1 4 dimℙ3
5. a. No. b. Yes, because dimℙ2 dimℙ3.
c.
1 0 0
0 1 0
0 0 1
0 0 0
Selected Answers to the Exercises 45
d. kerT zx, so it has no basis and nullityT 0. e. rangeT has basis −5 2x − x3, 3 − 5x 3x2 2x3, 6x − 5x2 and rankT 3. f. T is one-to-one but not onto. g. 3 0 dimℙ2.
6. b. No. c. Yes, because dimℙ2 dimℙ3.
d.
0 −5 −8
0 0 −6
0 1 0
0 0 4
e.
0 1 0
0 0 1
0 0 0
0 0 0
f. kerT has basis 1 and nullityT 1. g. rangeT has basis −5 x2,−8 − 6x 4x3 and rankT 2. h. T is neither one-to-one nor onto. i. 2 1 3 dimℙ2.
7. b. Yes, because dimℙ3 dimℙ2. c. No.
d.
6 −3 6 −21
−10 5 −10 35
2 −1 2 −7
e.
1 −1/2 1 −7/2
0 0 0 0
0 0 0 0
f. kerT has basis 1 2x,−1 x2, 7 2x3 and nullityT 3. g. rangeT has basis 6 − 10x 2x2 and rankT 1. h. T is neither one-to-one nor onto. i. 1 3 4 dimℙ3.
8. a. Yes, because dimℙ2 dimℙ1. b. No.
c. 1 0 2
7
0 1 − 27 7
d. kerT has basis 147 − 6x − 7x2 and nullityT 1 e. rangeT has basis x 3, 2x − 1 or 1, x; either basis is acceptable because rankT 2. f. T is not one-to-one but it is onto. g. 2 1 3 dimℙ2.
9. a. No. b. Yes, because dimℙ1 dimℙ2.
c.
1 0
0 1
0 0
d. kerT zx, so it has no basis and nullityT 0. e. rangeT Span5x2 − 6x − 9, 3x2 − x 9 and rankT 2. f. T is one-to-one but not onto. g. 2 0 2 dimℙ1.
10. a. Yes, because dimℙ2 dimℙ1. b. No.
c. 1 − 1
2 3 2
0 0 0
d. kerT has basis 2x 5,−2x2 2x − 3 and nullityT 2 e. rangeT has basis 3x − 7 and rankT 1.
46 Selected Answers to the Exercises
f. T is neither one-to-one nor onto. g. 1 2 3. 11. a. No. b. Yes, because dimℙ1 dimℙ2.
c.
1 5/7
0 0
0 0
d. kerT −7x 2, and nullityT 1. e. rangeT has basis 2x2 x 8 and rankT 1. f. T is neither one-to-one nor onto. g. 1 1 2 dimℙ1.
12. a. No. b. No.
c.
1 0 − 27 11
0 1 14 11
0 0 0
d. kerT has basis 27 − 14x 11x2, and nullityT 1. e. rangeT has basis 4 − x 5x2, 3 2x 12x2, and rankT 2. h. px 3 − 2x c2
11 27 − 14x 11x2 ( c2
11 can be replaced by c)
13. b. T1B,B/
4 −5 0
0 7 −10
0 1 10
0 0 2
, and T2B/,B
0 3 −10 0
0 0 6 −30
0 0 0 9
.
c. The codomain of the first is the same as the domain of the second, in either order.
d. T2 ∘ T1B,B 0 11 −130
0 6 0
0 0 18
and T1 ∘ T2B/,B/
0 12 −70 150
0 0 42 −300
0 0 6 60
0 0 0 18
14. b. T1B,B/
3 0 0
2 3 0
0 2 3
0 0 2
and T2B/,B//
1 −3 9 −27
0 1 4 12
0 0 2 −6
.
c. T2 ∘ T1B,B// −3 9 −27
2 11 36
0 4 −6
.
d. No, because the codomain of T2, which is 3, is not the domain of T1, which is ℙ2. The two spaces 3 and ℙ2 are both 3-dimensional, but the composition T1 ∘ T2 is still undefined. e. Yes, the matrix product T1B,B/ T2B/,B// is a well-defined 4 4 matrix. However, it is completely meaningless in this case.
15. a. No. b. Yes; domain ℙ2 and codomain ℙ1. c. 10x3 − 2x2 16x 11
Selected Answers to the Exercises 47
d. 36x − 167 e. 19 25 33
2 −12 62 16. a. Yes; domain ℙ2 and codomain ℙ2. b. Yes; domain ℙ1 and codomain ℙ1.
c. 35x2 − 127x − 11 d. 41x 7
e. 11 −14
13 3 g. 220x − 245 h. 1540x2 − 5525x − 295 i.
−13 9 −16
32 −1 14
49 −7 28
17. a. D 2 B
1 0
0 4 and D 3
B
−1 0
0 8 ; b. f //x 5e−x − 12e2x;
f ///x −5e−x − 24e2x
18. a. D 2 B
0 −2
2 0 and D 3
B
−2 −2
2 −2 ;
b. f //x 5e−x − 12e2x; f ///x −2ex sinx 14ex cosx
19. a. D 2 B
5 12
−12 5 and D 3
B
9 −46
46 9 ;
b. f //x −83e−3x sin2x − 105e−3x cos2x; f ///x 459e−3x sin2x 149e−3x cos2x
20. a. D 2 B
25 0
10 25 and D 3
B
125 0
75 125
b. f //x −50xe5x 155e5x; f ///x −250xe5x 725e5x
21. a. D 2 B
16 0 0
−16 16 0
2 −8 16
and D 3 B
−64 0 0
96 −64 0
−24 48 −64 b. f //x −80x2e−4x 112xe−4x − 138e−4x; f ///x 320x2e−4x − 608xe−4x 664e−4x
22. a. D 2 B
ln52 0 0
4 ln 5 ln52 0
2 2 ln 5 ln52 and
D 3 B
ln53 0 0
6ln52 ln53 0
6 ln 5 3ln52 ln53
b. f //x −4ln52x25x 9ln52 − 16 ln5 x5x −2ln52 18 ln5 − 8 5x;
f ///x −4ln53x25x 9ln53 − 24ln52 x5x
−2ln53 27ln52 − 24 ln5 5x
48 Selected Answers to the Exercises
23. a. D 2 B
−4 0 0 0
0 −4 0 0
0 −4 −4 0
4 0 0 −4
and D 3 B
0 8 0 0
−8 0 0 0
−12 0 0 8
0 −12 −8 0 b. f //x −16x sin2x − 36x cos2x − 16 sin2x − 16 cos2x;
f ///x 72x sin2x − 32x cos2x 16 sin2x − 68 cos2x
24. a. D 2 B
a2 − b2 −2ab
2ab a2 − b2 and D 3
B
a3 − 3ab2 b3 − 3a2b
−b3 3a2b a3 − 3ab2
3.8 Exercises
1. b. TB,B/
1 −3 9
1 5 25
0 1 4
. c. TB,B/ −1
− 5 16
21 16
− 15 2
− 1 4
1 4
−1 1
16 − 1
16 1 2
d.
px 9 − 7x 5x2.
2. b. TB,B/
1 −4 16 −64
1 1 1 1
1 3 9 27
0 1 −2 3
. c.
− 3 25
33 25
− 1 5 − 6
5
19 175
− 23 100
17 140
13 10
1 35
− 1 10
1 14
0
− 3 175
1 100
1 140
− 1 10
d. px −11 7x − 5x2 2x3. 3. a. 5x2 − 9x 14 b. −3x2 4x 7 4. a. 9x2 − 5x 17 b. −8x2 − 19x 23 5. a. −4x2 9x − 3 b. 15x2 − 8x − 11 6. a. −5x3 8x2 − 3x 11 b. −13x2 7x 11 7. a. −4x3 12x2 19x − 7 b. 17x3 − 5x2 12x 8 8. a. −9x3 13x2 − 5x 11 b. 4x3 − 15x 8 9. a. 9x3 7x2 − 11 b. 11x3 − 18x 9 10. a. 2
3 x3 − 9x2 − 11x 17 b. −12x3 7
4 x2 9x − 3
11. a. DB −1 1
13 −3 2
−2 −3 b. 7e−3x sin2x − 5e−3x cos2x C.
12. a. DB −1 1
25 5 0
−1 5 b. 3xe5x 8e5x C.
13. a. DB −1 1
32
−8 0 0
−4 −8 0
−1 −2 −8
b. 4x2e−4x − 9xe−4x − 3e−4x C.
Selected Answers to the Exercises 49
14. a. DB −1
1 ln 5
0 0
− 2 ln 52
1 ln 5
0
2 ln 53
− 1 ln 52
1 ln 5
b. 7 ln 5
x2 5x − 14 ln 52
4 ln 5
x 5x 14 ln 53
4 ln 52
9 ln 5
5x C.
15. a. DB −1 1
4
0 2 0 0
−2 0 0 0
1 0 0 2
0 1 −2 0
;
b. 3x sin2x − 7x cos2x − 5 sin2x 6 cos2x C
16. a. DB −1 1
k2 m2 k m
−m k
b. k k2 m2
ekx sinmx − m k2 m2
ekx cosmx C and m
k2 m2 ekx sinmx k
k2 m2 ekx cosmx C.
17. fx −2x2e−3x 8xe−3x 3e−3x 18. a. W ℙ2 (use the standard basis) b. T 3I3 5D − 2D2
c. TB
3 5 −4
0 3 10
0 0 3
d. TB −1 1
27
9 −15 62
0 9 −30
0 0 9
e. 1 3 2 − 7x 5x2
19. a. W ℙ3 (use the standard basis) b. T 3I4 5D − 2D2
c. TB
3 5 −4 0
0 3 10 −12
0 0 3 15
0 0 0 3
d. TB −1 1
27
9 −15 62 −370
0 9 −30 186
0 0 9 −45
0 0 0 9
e. − 3511 1752x − 747x2 162x3 20. a. W SpanB, B sinx, cosx b. T −7IW 8D 3D2
c. TB −10 −8
8 −10 d. TB
−1 1 82
−5 4
−4 −5 e. −12 sinx 7 cosx
21. a. W SpanB, B sinx, cosx b. T 8IW 3D − 4D2 − 2D3
c. TB 12 −5
5 12 d. TB
−1 1 169
12 5
−5 12 e. 5 sinx 7 cosx
22. a. W SpanB, B sin2x, cos2x b. T −7IW 8D 3D2
c. TB −19 −16
16 −19 d. TB
−1 1 617
−19 16
−16 −19 e. −5 sin2x − 14 cos2x
23. a. W SpanB, B sin2x, cos2x b. T 8IW 3D − 4D2 − 2D3
50 Selected Answers to the Exercises
c. TB 24 −22
22 24 d. TB
−1 1 530
12 11
−11 12 e. 3 sin2x − 8 cos2x
24. a. W SpanB, B e−3x sin2x, e−3x cos2x b. T 4IW 5D − 9D2
c. TB −56 −118
118 −56 d. TB
−1 1 8530
−28 59
−59 −28 e.
17e−3x sin2x 11e−3x cos2x 25. a. W SpanB, B e−3x sin2x, e−3x cos2x b. T −6IW 2D 7D2 3D3
c. TB 50 −58
58 50 d. TB
−1 1 2932
25 29
−29 25 e.
5e−3x sin2x 2e−3x cos2x 26. a. W SpanB, B xe5x, e5x b. T 4IW − 9D 2D2
c. TB 9 0
11 9 d. TB
−1 1 81
9 0
−11 9 e. 4xe5x − 7e5x
27. a. W SpanB, B xe5x, e5x b. T 2IW − 7D − 3D2 4D3
c. TB 64 0
77 64 d. TB
−1 1 4096
64 0
−77 64 e. −9xe5x 13e5x
28. a. W SpanB, B x2e−4x, xe−4x, e−4x b. T 8IW 11D 3D2
c. TB
12 0 0
−26 12 0
6 −13 12
d. TB −1 1
864
72 0 0
156 72 0
133 78 72
e. 3x2e−4x 7xe−4x 5e−4x 29. a. W SpanB, B x2e−4x, xe−4x, e−4x b. T 11IW − 8D 4D2 3D3
c. TB
−85 0 0
208 −85 0
−64 104 −85
d. TB −1 −1
614125
7225 0 0
17680 7225 0
16192 8840 7225
e. 2x2e−4x 9xe−4x 7e−4x 30. a. W SpanB, B sinh3x, cosh3x b. T −8IW 9D 4D2
c. TB 28 27
27 28 d. TB
−1 1 55
28 −27
−27 28 e. −4 sinh3x 5 cosh3x.
31. a. W SpanB, B x sin2x, x cos2x, sin2x, cos2x b. T 6IW 4D 3D2
c. TB
−6 −8 0 0
8 −6 0 0
4 −12 −6 −8
12 4 8 −6
d. TB −1 1
1250
−75 100 0 0
−100 −75 0 0
158 6 −75 100
−6 158 −100 −75 e. −7x sin2x 5x cos2x 4 sin2x − 6 cos2x
32. d. False.
Selected Answers to the Exercises 51
33. −8 3x − 4x2 x3/2 34. −3 19x − 3
2 x2
35. 13 − 11x 8x2
36. 9 5x − 3 2
x2 2x3
37. 3 4x − 7x3 38. 9 − 3x − 8x2 5x3 − 7x4 39. a. It is a diagonal matrix where none of the diagonal entries is 0. b. ⟨−147, 559/2,−632
c. T−1 B/,B
Diag1/3, 2,−1/5 d. − 92 3
74 5
x 2 5
x2
40. a. It is a triangular matrix where none of the diagonal entries is 0. b. ⟨−26, 109/3,−175/3
c. T−1 B/,B
− 1 2 − 15
2 − 37
2 0 3 6
0 0 −1
d. 79 44x 2x2
41. a. 1 20
−5 5 5
8 4 −4
1 3 7
b. 34 − 29x 7x2.
52 Selected Answers to the Exercises
Chapter Four Exercises
4.1 Exercises 1. a. ⟨1,−1,−12, 6,⟨11,−16, 13, 1,⟨1, 1,−16, 10; dimV ∨ W 3.
b. ⟨5,−7,−2, 4; dimV ∩ W 1, c. ⟨5,−7,−2, 4 35 ⟨1,−1,−12, 6
2 5 ⟨11,−16, 13, 1, and
⟨5,−7,−2, 4 1 3 ⟨1, 1,−16, 10 2
3 ⟨7,−11, 5, 1. d. 3 2 2 − 1.
2. a. ⟨3, 5,−2, 4,⟨1, 2, 7,−3,⟨0, 2, 1,−5,⟨2,−3, 1, 6; dimV ∨ W 4 i.e. V ∨ W 4. b. dimV ∩ W 0, so it has no basis. 4 2 2 − 0, verifying d..
3. a. ⟨−3,−2, 7,−4,⟨−2, 13,−12,−2,⟨−2, 3,−5, 1,⟨−3,−5, 6,−11; dimV ∨ W 4 i.e. V ∨ W 4. b. ⟨−26,−17, 14, 0,⟨3,−8, 0, 7; dimV ∩ W 2 c. ⟨−26,−17, 14, 0 4⟨−3,−2, 7,−4 − 3⟨−2, 13,−12,−2 10⟨−2, 3,−5, 1, and ⟨3,−8, 0, 7 −⟨−3,−2, 7,−4 − ⟨−2, 13,−12,−2 ⟨−2, 3,−5, 1; ⟨−26,−17, 14, 0 4⟨−3,−5, 6,−11 − 3⟨−1, 16,−8, 8 − 17⟨1,−3, 2,−4, and ⟨3,−8, 0, 7 −⟨−3,−5, 6,−11 − ⟨−1, 16,−8, 8 − ⟨1,−3, 2,−4. d. 4 3 3 − 2.
4. a. ⟨−3, 4,−1, 4, 6,⟨−6, 8, 5, 15,−13,⟨1,−2, 0,−5, 3,⟨1, 3,−2, 7, 2; dimV ∨ W 4. b. ⟨3,−2,−5, 0, 4; dimV ∩ W 1. c. ⟨3,−2,−5, 0, 4 0⟨−3, 4,−1, 4, 6 − ⟨−6, 8, 5, 15,−13 − 3⟨1,−2, 0,−5, 3, and ⟨3,−2,−5, 0, 4 −⟨1, 3,−2, 7, 2 − ⟨−4,−1, 7,−7,−6. d. 4 3 2 − 1.
5. a. ⟨−1, 7, 5,−6, 6,⟨−1,−8, 2,−4, 2,⟨1, 0, 3,−4, 3,⟨5, 3,−2, 7,−4,⟨−6, 9,−2, 0, 0; dimV ∨ W 5 i.e. V ∨ W 5. b. ⟨−17, 31,−3, 0, 4,⟨−3, 7,−1, 2, 0; dimV ∩ W 2. c. ⟨−17, 31,−3, 0, 4 3⟨−1, 7, 5,−6, 6 − 2⟨−1,−8, 2,−4, 2 − 6⟨1, 0, 3,−4, 3 − 2⟨5, 3,−2, 7,−4 and ⟨−3, 7,−1, 2, 0 ⟨−1, 7, 5,−6, 6 − 2⟨1, 0, 3,−4, 3; ⟨−17, 31,−3, 0, 4 3⟨−6, 9,−2, 0, 0 − 2⟨−5, 1,−3,−3,−2 3⟨−3, 2,−1,−2, 0, and ⟨−3, 7,−1, 2, 0 ⟨−6, 9,−2, 0, 0 − ⟨−3, 2,−1,−2, 0. d. 5 4 3 − 2.
6. a. 6 − x 2x2 10x3, 11 − 3x 6x2 2x3, 3 17x 5x2 4x3; dimV ∨ W 3 b. −3 x − 2x2 2x3; dimV ∩ W 1. c. −3 x − 2x2 2x3 2
7 6 − x 2x2 10x3 − 3
7 11 − 3x 6x2 2x3, and
− 3 x − 2x2 2x3 23 17x 5x2 4x3 − 33 11x 4x2 2x3 d. 3 2 2 − 1.
7. a. 2 5x − 10x2 5x3, 6 − 7x − 4x2 x3,−8 14x − 16x2 3x3; dimV ∨ W 4, i.e. V ∨ W ℙ3. b. 4 − x − 7x2 3x3; dimV ∩ W 1. c. 4 − x − 7x2 3x3 1
2 2 5x − 10x2 5x3 1
2 6 − 7x − 4x2 x3, and
4 − x − 7x2 3x3 3 5 2 3x − 19x2 13x3 2
5 7 − 7x 11x2 − 12x3 d.
4 3 2 − 1. 8. a. −3 − 2x 4x2 x4, 6 − 3x2 5x3 − 5x4,−7 − 7x 8x2 2x3 8x4,
− 5 − 2x 7x2 x3 − x4, 1 − 6x 3x2 − 2x3 − 4x4; dimV ∨ W 5, i.e. V ∨ W ℙ4. b. 5008 9057x − 12636x2, 28 − 33x 52x3, 18 15x − 52x4; dimV ∩ W 3. c. 56 − x − 52x2 30−3 − 2x 4x2 x4 56 − 3x2 5x3 − 5x4 −
3−7 − 7x 8x2 2x3 8x4 − 19−5 − 2x 7x2 x3 − x4;
Selected Answers to the Exercises 53
28 − 33x 52x3 2−3 − 2x 4x2 x4 96 − 3x2 5x3 − 5x4 5−7 − 7x 8x2 2x3 8x4 − 3−5 − 2x 7x2 x3 − x4, and 18 15x − 52x4 18−3 − 2x 4x2 x4 36 − 3x2 5x3 − 5x4 − 7−7 − 7x 8x2 2x3 8x4 − −5 − 2x 7x2 x3 − x4; 56 − x − 52x2 −261 − 6x 3x2 − 2x3 − 4x4 55 − 14x 7x2 x3 − 12x4 − 3−9x 3x2 2x4 − 19−3 6x 3x3 2x4; 28 − 33x 52x3 −261 − 6x 3x2 − 2x3 − 4x4 95 − 14x 7x2 x3 − 12x4 5−9x 3x2 2x4 − 3−3 6x 3x3 2x4;
18 15x − 52x4 35 − 14x 7x2 x3 − 12x4 − 7−9x 3x2 2x4 − −3 6x 3x3 2x d. 5 4 4 − 3.
11. 6 ≤ dimV ∩ W ≤ 8. 13. W must be a subspace of V. 14. W must be a subspace of V. 15. V ∩ W W, or V ∨ W V.
4.2 Exercises
1. a. ⟨1, 0, 4,⟨0, 1,−7 b. ⟨3, 5, 4,−1,⟨2, 3, 2,−1 c. 17 −28
−28 50 d.
25 33
14 33
14 33
17 66
2. a. ⟨1,−3, 0,⟨0, 0, 1 b. ⟨2,−3,−4, 5,⟨−7,−1, 9, 3 c. 10 0
0 1 d.
1 10
0
0 1
3. a. ⟨1, 0, 0,⟨0, 1, 0,⟨0, 0, 1 b. ⟨2,−3,−4, 5,⟨−6, 9, 12,−5, ,⟨−7,−1, 9, 3 c. I3, with inverse d. I3. Note, though that this is not the identity transformation.
4. a. ⟨1, 0,−2,−2,⟨0, 1, 2, 1 b. ⟨3, 2,−2,⟨5, 3,−1 c. 9 −6
−6 6 d.
1 3
1 3
1 3
1 2
5. a. ⟨1, 0,−2,⟨0, 1, 1 b. ⟨3, 2,−2,⟨5, 3,−1 c. 5 −2
−2 2 d.
1 3
1 3
1 3
5 6
6. a. ⟨1, 5, 0, 4,⟨0, 0, 1,−3 b. ⟨2, 3,−4,⟨5, 7,−9 c. 42 −12
−12 10 d.
5 138
1 23
1 23
7 46
7. a. ⟨1, 5, 0, 0,⟨0, 0, 1, 0,⟨0, 0, 0, 1 b. ⟨2, 3,−4,⟨5, 7,−9,⟨−7,−9, 8
c.
26 0 0
0 1 0
0 0 1
d.
1 26
0 0
0 1 0
0 0 1
8. a. ⟨1, 4, 0,−5, 2,⟨0, 0, 1, 6,−3 b. ⟨−5, 3, 2,−4,⟨−4,−2, 3, 1
c. 46 −36
−36 46 d.
23 410
9 205
9 205
23 410
9. a. ⟨1, 4, 0,−5,⟨0, 0, 1, 6 b. ⟨−5, 3, 2,−4,⟨−4,−2, 3, 1
54 Selected Answers to the Exercises
c. 42 −30
−30 37 d.
37 654
5 109
5 109
7 109
10. a. ⟨1,−2, 6, 0,−4,⟨0, 0, 0, 1, 5 b. ⟨−5, 2,−3, 4,⟨−3,−3, 2,−5
c. 57 −20
−20 26 d.
13 541
10 541
10 541
57 1082
11. a. ⟨1,−3, 0, 0, 2,⟨0, 0, 1, 0,−4,⟨0, 0, 0, 1, 7 b. ⟨−2, 3, 4,−5,⟨−3, 7,−1, 2,⟨−5, 4,−2, 3
c.
14 −8 14
−8 17 −28
14 −28 50
d.
33 332
1 83 − 7
332
1 83
63 83
35 83
− 7 332
35 83
87 332
12. a. ⟨1,−3, 0, 0, 0,⟨0, 0, 1, 0, 0,⟨0, 0, 0, 1, 0,⟨0, 0, 0, 0, 1 b. ⟨−2, 3, 4,−5,⟨−3, 7,−1, 2,⟨−5, 4,−2, 3,⟨−7, 6,−2, 3
c.
10 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
d.
1 10
0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
13. a. ⟨1, 5, 0,−2,⟨0, 0, 1, 6 b. ⟨2,−4, 3, 5,−6,⟨−1, 1, 1,−3, 2
c. 30 −12
−12 37 d.
37 966
2 161
2 161
5 161
14. a. ⟨1, 5, 0, 0,⟨0, 0, 1, 0,⟨0, 0, 0, 1 b. ⟨2,−4, 3, 5,−6,⟨−1, 1, 1,−3, 2,⟨−9, 14, 0,−28, 24
c.
26 0 0
0 1 0
0 0 1
d.
1 26
0 0
0 1 0
0 0 1
15. a. ⟨1, 0, 0,−2,⟨0, 1, 0, 3,⟨0, 0, 1,−5 b. ⟨−2, 5, 1,−2,−1,⟨1,−1, 1,−2, 1,⟨1,−1,−1, 1, 1
c.
5 −6 10
−6 10 −15
10 −15 26
d.
35 39
2 13 − 10
39
2 13
10 13
5 13
− 10 39
5 13
14 39
16. a. The standard basis for 4 b. The four columns of T. c. I3 d. I3. Again, this is not the identity transformation.
4.3 Exercises 1. a. ⟨−5, 1, 12, 11 b. ⟨1, 1, 0,⟨−4, 7, 1 2. a. ⟨−7,−1, 9, 3,⟨2,−3,−4, 5 (scaled down) b. ⟨2, 0, 1,⟨3, 1, 0,⟨0, 0, 1 3. a. ⟨−7,−1, 9, 7,⟨2,−3,−4, 3 (scaled down) b. ⟨1, 0, 1,⟨3, 1, 0 (scaled up) 4. a. ⟨3, 2,−2,⟨5, 2, 6 (scaled down) b. ⟨−1, 1, 0, 0,⟨2, 1, 0, 0,⟨2,−2, 1, 0,⟨2,−1, 0, 1
Selected Answers to the Exercises 55
5. a. ⟨5, 4,−8,⟨3, 2,−2 b. ⟨−1, 1, 0,⟨2,−1, 1,⟨1,−1, 1 6. a. ⟨4, 7,−10,⟨13, 17,−21 b. ⟨−1, 0, 1, 0,⟨−5, 1, 0, 0,⟨−4, 0, 3, 1 7. a. ⟨8, 10,−9,⟨−9,−14, 16,⟨23, 38,−65 or simply i, j, k b.
⟨−1, 0, 1, 0,⟨−15, 0, 5,−1,⟨−5, 1, 0, 0 8. a. ⟨−43, 105,−8,−110,⟨−9, 1, 5,−3 (scaled down) b.
⟨−1, 0, 1, 0, 0,⟨−4, 1, 0, 0,⟨5, 0,−6, 1, 0,⟨−2, 0, 3, 0, 1 9. a. ⟨−88, 22, 45,−41,⟨−92, 350,−57,−355 b. ⟨1, 0, 1, 0,⟨−4, 1, 0, 0,⟨5, 0,−6, 1 10. a. ⟨192, 3, 43,−13 (scaled down) b.
⟨1, 0, 0,−1, 0,⟨2, 1, 0, 0, 0,⟨−6, 0, 1, 0, 0,⟨4, 0, 0,−5, 1 11. a. ⟨15, 13, 45,−58 b. ⟨3, 1, 0, 0, 0,⟨−2, 0, 4,−7, 1 12. a. ⟨−88, 97, 7,−1,⟨4,−40,−79, 100
b. − 1 10
, 0, 1 5
,− 7 20
, 1 20
, − 1 10
, 0 1 5
,− 17 20
, 11 20
, 9 10
, 0, 1 5
, 143 20
,− 109 20
,⟨3, 1, 0, 0, 0 13. a. ⟨12,−22, 13, 31,−34,⟨−61, 85, 1,−171, 146 b. ⟨2, 0, 1, 0,⟨−5, 1, 0, 0,⟨2, 0,−6, 1 14. a. ⟨−21, 31, 3,−65, 54,⟨−53, 78, 8,−164, 136 b.
⟨2, 0,−6, 1,⟨− 3 4
, 0, 3 4
,− 1 4 , − 1
8 , 0,− 11
8 ,− 1
8 ,⟨−5, 1, 0, 0 . Since this basis has 4
elements, the preimage is all of 4, so any basis for 4 is also a correct answer, including the standard basis.
15. a. ⟨−6, 21,−11, 12,−1,⟨7,−25, 19,−23, 1,⟨−2, 14,−12, 13, 2 b. ⟨0, 2,−1, 0, 0,⟨2,−3, 5, 1
4.4 Exercises 1. Yes. 2. No. 3. Yes. 4. No. 5. Yes. 6. No. 7. Yes. 8. Yes. 9. No. 10. Yes. 11. x0 −21 and z0 14. 12. x0 30, y0 −45, and z0 18. 13. a. ⟨3,−1, 2, 0, e1, e2, e4 b. e1, e2, e4 c. 3; d. 3 4 − 1 14. a. ⟨3, 5, 2,−2,⟨−2, 1, 2,−2, e1, e3 b. e1, e3 c. 2; d. 2 4 − 2 15. a. ⟨3, 0,−2, 0, 7, e1, e2, e3, e4 b. e1, e2, e3, e4 c. 4; d. 4 5 − 1 16. a. ⟨2, 0, 7, 3, 0,⟨0, 5,−14,−6, 0, e1, e3, e5 b. e1, e3, e5 c. 3; d. 3 5 − 2 17. a. ⟨4,−3, 0, 0, 5,⟨2,−3, 0, 0, 5,⟨2, 1, 0, 0, 5, e3, e4 b. e3, e4 c. 2 d. 2 5 − 3
4.5 Exercises 1. a. ⟨3, 5, 4,−1,⟨2, 3, 2,−1 b. ⟨−4, 7, 1 c. e1 kerT, e2 kerT
d. Te1 kerT c1;
Te2 kerT c2
2. a. ⟨2,−3,−4, 5,⟨−7,−1, 9, 3 b. ⟨3, 1, 0 c. e1 kerT, e3 kerT d. Te1 kerT c1;
Te3 kerT c3
3. a. ⟨3, 2,−2,⟨5, 3,−1 b. ⟨2,−2, 1, 0,⟨2,−1, 0, 1 c. e1 kerT, e2 kerT d. Te1 kerT c1;
Te2 kerT c2
4. a. ⟨3, 2,−2,⟨5, 3,−1 b. ⟨2, 1, 0 c. e1 kerT, e2 kerT d. Te1 kerT c1;
Te2 kerT c2
5. a. ⟨2, 3,−4,⟨5, 7,−9 b. ⟨−5, 1, 0, 0,⟨−4, 3, 0, 1 c. e1 kerT, e3 kerT d. Te1 kerT c1;
Te3 kerT c3
6. a. ⟨2, 3,−4,⟨5, 7,−9,⟨−7,−9, 8 b. ⟨−5, 1, 0, 0 c. e1 kerT, e3 kerT, e4 kerT d. Te1 kerT c1;
Te3 kerT c3;
Te4 kerT c4
7. a. ⟨−5, 3, 2,−4,⟨−4,−2, 3, 1 b. ⟨−4, 1, 0, 0, 0,⟨5, 0,−6, 1, 0,⟨−2, 0, 3, 0, 1 c. e1 kerT, e3 kerT d.
Te1 kerT c1;
Te3 kerT c3
56 Selected Answers to the Exercises
8. a. ⟨−5, 3, 2,−4,⟨−4,−2, 3, 1 b. ⟨−4, 1, 0, 0, 0,⟨5, 0,−6, 1, 0 c. e1 kerT, e3 kerT d.
Te1 kerT c1;
Te3 kerT c3
9. a. ⟨−5, 2,−3, 4,⟨−3,−3, 2, 5 b. ⟨2, 1, 0, 0, 0,⟨−6, 0, 1, 0, 0,⟨4, 0, 0,−5, 1 c. e1 kerT, e4 kerT d.
Te1 kerT c1;
Te4 kerT c4
10. a. ⟨−2, 3, 4,−5,⟨−3, 7,−1, 2,⟨−5, 4,−2, 3 b. ⟨3, 1, 0, 0, 0,⟨−2, 0, 4,−7, 1 c. e1 kerT, e3 kerT, e4 kerT d.
Te1 kerT c1;
Te3 kerT c3;
Te4 kerT c4 11. a. ⟨−2, 3, 4,−5,⟨−3, 7,−1, 2,⟨−5, 4,−2, 3,⟨−7, 6,−2, 3 b. ⟨3, 1, 0, 0, 0
c. e1 kerT, e3 kerT, e4 kerT, e5 kerT d. Te1 kerT c1;
Te3 kerT c3;
Te4 kerT c4;
Te5 kerT c5
12. a. ⟨2,−4, 3, 5,−6,⟨−1, 1, 1,−3, 2 b. ⟨−5, 1, 0, 0,⟨2, 0,−6, 1 c. e1 kerT, e3 kerT d.
Te1 kerT c1;
Te3 kerT c3
13. a. ⟨2,−4, 3, 5,−6,⟨−1, 1, 1,−3, 2,⟨−9, 14, 0,−28, 24 b. ⟨−5, 1, 0, 0 c. e1 kerT, e3 kerT, e4 kerT d. Te1 kerT c1;
Te3 kerT c3;
Te4 kerT c4
14. a. ⟨−2, 5, 1,−2,−1,⟨1,−1, 1,−2, 1,⟨1,−1,−1, 1, 1 b. ⟨2,−3, 5, 1 c. e1 kerT, e2 kerT, e3 kerT d. Te1 kerT c1;
Te2 kerT c2;
Te3 kerT c3
15. a. ⟨⟨−2,−1, 1 U b. e2 W c. ⟨−2,−1, 1 U, e2 U d. e2 W/U e. Te2 U W/U e2 W
16. a. ⟨1, 1, 1, 2 U b. e1 W, e3 W c. ⟨1, 1, 1, 2 U, e1 U, e3 U d. e1 W/U, e3 W/U e.
Te1 U W/U e1 W;
Te3 U W/U e3 W;
17. a. ⟨1, 1, 1, 2 U,⟨3,−1, 1, 2 U b. e3 W c. ⟨1, 1, 1, 2 U,⟨3,−1, 1, 2 U, e3 U d. e3 W/U e.
Te3 U W/U e3 W
18. a. ⟨3,−1, 1, 2 U b. e3 W c. ⟨3,−1, 1, 2 U, e3 U d. e3 W/U e. Te3 U W/U e3 W
19. a. ⟨1, 1, 1, 2,−3 U,⟨3,−1, 1, 2,−3 U b. e3 W, e4 W c. ⟨1, 1, 1, 2,−3 U,⟨3,−1, 1, 2,−3 U, e3 U, e4 U d. e3 W/U, e4 W/U e.
Te3 U W/U e3 W;
Te4 U W/U e4 W
20. a. ⟨3,−1, 1, 2,−3 U b. e3 W, e4 W c. ⟨3,−1, 1, 2,−3 U, e3 U, e4 U d. e3 W/U, e4 W/U e.
Te3 U W/U e3 W;
Te4 U W/U e4 W
21. a. ⟨3,−1, 1, 2,−3 U,⟨3,−1, 1,−1,−3 U b. e3 W c. ⟨3,−1, 1, 2,−3 U,⟨3,−1, 1,−1,−3 U, e3 U d. e3 W/U e. Te3 U W/U e3 W
22. a. ⟨1,−1,−12, 6,⟨11,−16, 13, 1,⟨1, 1,−16, 10 b. ⟨5,−7,−2, 4 c. ⟨1,−1,−12, 6 W,⟨1, 1,−16, 10 W d. ⟨1,−1,−12, 6 V ∩ W,⟨1, 1,−16, 10 V ∩ W e. ⟨1,−1,−12, 6 V,⟨1, 1,−16, 10 V f. ⟨1,−1,−12, 6 V ∩ W,⟨1, 1,−16, 10 V ∩ W g. T 1⟨1,−1,−12, 6 W ⟨1,−1,−12, 6 V ∩ W;
T 1⟨1, 1,−16, 10 W ⟨1, 1,−16, 10 V ∩ W; h. T 2⟨1,−1,−12, 6 V ⟨1,−1,−12, 6 V ∩ W;
T 2⟨1, 1,−16, 10 V ⟨1, 1,−16, 10 V ∩ W 23. a. ⟨3, 5,−2, 4,⟨1, 2, 7,−3,⟨0, 2, 1,−5,⟨2,−3, 1, 6 b. dimV ∩ W 0, so it has no
Selected Answers to the Exercises 57
basis. c. ⟨3, 5,−2, 4 W,⟨1, 2, 7,−3 W d. ⟨3, 5,−2, 4 04 ,⟨1, 2, 7,−3 04 e. ⟨0, 2, 1,−5 V,⟨2,−3, 1, 6 V f. ⟨0, 2, 1,−5 04 ,⟨2,−3, 1, 6 04 g. T 1⟨3, 5,−2, 4 W ⟨3, 5,−2, 4 04 ;
T 1⟨1, 2, 7,−3 W ⟨1, 2, 7,−3 04 ;
h. T 2⟨0, 2, 1,−5 V ⟨0, 2, 1,−5 04 ;
T 2⟨2,−3, 1, 6 V ⟨2,−3, 1, 6 04
24. a. ⟨−3,−2, 7,−4,⟨−2, 13,−12,−2,⟨−2, 3,−5, 1,⟨−3,−5, 6,−11 b. ⟨−26,−17, 14, 0,⟨3,−8, 0, 7 c. ⟨−3,−2, 7,−4 W d. ⟨−3,−2, 7,−4 V ∩ W e. ⟨−3,−5, 6,−11 V f. ⟨−3,−5, 6,−11 V ∩ W g. T 1⟨−3,−2, 7,−4 W ⟨−3,−2, 7,−4 V ∩ W h.
T 2⟨−3,−5, 6,−11 V ⟨−3,−5, 6,−11 V ∩ W 25. a. ⟨−3, 4,−1, 4, 6,⟨−6, 8, 5, 15,−13,⟨1,−2, 0,−5, 3,⟨1, 3,−2, 7, 2 b. ⟨3,−2,−5, 0, 4
c. ⟨−3, 4,−1, 4, 6 W,⟨−6, 8, 5, 15,−13 W d. ⟨−3, 4,−1, 4, 6 V ∩ W,⟨−6, 8, 5, 15,−13 V ∩ W e. ⟨1, 3,−2, 7, 2 V f. ⟨1, 3,−2, 7, 2 V ∩ W g. T 1⟨−3, 4,−1, 4, 6 W ⟨−3, 4,−1, 4, 6 V ∩ W;
T 1⟨−6, 8, 5, 15,−13 W ⟨−6, 8, 5, 15,−13 V ∩ W; h. T 2⟨1, 3,−2, 7, 2 V ⟨1, 3,−2, 7, 2 V ∩ W
26. a. ⟨−1, 7, 5,−6, 6,⟨−1,−8, 2,−4, 2,⟨1, 0, 3,−4, 3,⟨5, 3,−2, 7,−4,⟨−6, 9,−2, 0, 0 b. ⟨−17, 31,−3, 0, 4,⟨−3, 7,−1, 2, 0 c. ⟨−1, 7, 5,−6, 6 W,⟨−1,−8, 2,−4, 2 W d. ⟨−1, 7, 5,−6, 6 V ∩ W,⟨−1,−8, 2,−4, 2 V ∩ W e. ⟨−6, 9,−2, 0, 0 V f. ⟨−6, 9,−2, 0, 0 V ∩ W g. T 1⟨−1, 7, 5,−6, 6 W ⟨−1, 7, 5,−6, 6 V ∩ W;
T 1⟨−1,−8, 2,−4, 2 W ⟨−1,−8, 2
h. T 2⟨−6, 9,−2, 0, 0 V ⟨−6, 9,−2, 0, 0 V ∩ W
58 Selected Answers to the Exercises
Chapter Five Exercises
5.1 Exercises 1. 3; 2. − 23; 3. − 11/3; 4. −5 3 ; 5. 4 ln 2 7 ln 3; 6. 1/2; 7. –47 8. 148; 9.
27/8; 10. −29/3; 11. 1800; 12. −70 ln 2 − 49 ln 5 13. a. ab; b. it is invertible if and only if both a and b are non-zero; c.
1 ab
b 0
0 a
1 a 0
0 1 b
.
14. a. a2 b2; b. it is invertible if and only if either a or b is non-zero; c.
1 a2 b2
a −b
b a .
15. a. a2 − b2; b. it is invertible if and only if a ≠ b; c. 1 a2 − b2
a −b
−b a .
16. a. 2ab; b. it is invertible if and only if both a and b are non-zero; c.
1 2ab
b −a
b a
1 2a − 1
2b
1 2a
1 2b
.
17. a. b − a; b. it is invertible if and only if a ≠ b; c. 1 b − a
b −a
−1 1 .
18. a. 2ea; b. it is always invertible; c. e −a
2 e−a −e−a
e2a e2a 1
2 e−2a −e−2a
ea ea .
19. a. 1; b. it is always invertible; c. cos sin
−sin cos .
20. a. 1; b. it is always invertible; c. cosha −sinha −sinha cosha
21. a. sin ; b. it is invertible if and only if ≠ n, where n is an integer; c.
1 sin
sin sin
−cos cos
22. 2, 4, 1, 3; both have 3 inversions. 23. 5, 3, 2, 4, 1; both have 8 inversions. Notice that −1.
24. 5, 3, 6, 1, 4, 2; both have 10 inversions. 25. 6, 4, 2, 7, 5, 1, 3; both have 14 inversions. 26. 5, 7, 3, 8, 4, 1, 6, 2; both have 18 inversions. 27. 2, 1, 4, 3; 2 inversions. 28. 2, 3, 5, 4, 1; 5 inversions. 29. 4, 1, 2, 5, 6, 3; 5 inversions. 30. 6, 3, 4, 2, 5, 1, 7; 11
inversions. 31. 6, 2, 3, 5, 1, 7, 8, 4; 11 inversions. 32. a. 0; b. 0; c. −1. 33. a. 0; b. 0; c. −1.
Selected Answers to the Exercises 59
34. c − ac − bb − a (other factorizations are possible, up to 1 35. the permutation n, n − 1, . . . , 3, 2, 1 will have n − 1 3 2 1 n − 1n/2
inversions.
5.2 Exercises 1. − 2. 3. − 4. 5. − 6. − 7. missing 2; 8. missing 4; − 9.
missing 3; 10. missing 5; 11. missing 2 and 5; − 12. missing 7 and 4; − 13. 0; column 2 is
all zeroes. 14. 0; the third row is 4 times the first 15. −30; the matrix is upper triangular 16. 7/5; the matrix is upper triangular; 17. −2640; the matrix is lower triangular 18. 60; the matrix is upper triangular; 19. 3780; the matrix is upper triangular 20. −1/4; the matrix is lower triangular. 21. −560 22. 360 23. 720 24. −7/2 25. a. − 5; b. 5; c. −20 d. 1/3 26. a. 42; b. 20 c. 10800 d. 40 27. a. 9 b. 270 c. −252 d. 0 28. a. −70; b. 6 c. 480 d. −588 29. −321; 30. 93; 31. 2981 32. 403 33. 863; 34. − 1779; 35. − 182 36. −448 37. −439; 38. 9730; 39. − 29700 40. 214295 41. a. detA 76; detB 345;
detC 421
5.3 Exercises
1. a. detA −34 and detB 46. b. AB 38 36
16 −26 and detAB −1564.
c. −1564 −3446 d. A B 11 4
4 5 and detA B 39.
e. 39 ≠ −34 46. f. 3B 18 −12
3 21 and det3B 414. g. det3B 9detB.
2. a. 7
−1 3 −4
2 −8 3
6 5 7
− −2 2 −3 2
−1 3 −4
2 −8 3
b. first determinant is −149 and the other is
−27; c. −1097
3. a. −6 −3 7 −2
3 6 4
−8 −2 3
− −4 5 −3 −2
−1 3 4
2 −8 3
b. first determinant is −449 and the other
is 168; c. 3366
4. a.
4 −2 3 8
9 0 17 28
2 3 −2 3
−3 0 9 −5
b.
6 1 1 11
9 0 17 28
2 3 −2 3
−3 0 9 −5
60 Selected Answers to the Exercises
c.
6 1 1 11
9 0 17 28
−16 0 −5 −30
−3 0 9 −5
d. − 9 17 28
−16 −5 −30
−3 9 −5
e. 1627 5. a. −219 b. −180 6. a. −255 b. 2452 7. a. −511 b. −1578 8. a. 56 b. −43 9. a. −42 b. −686 14. 140 16. a. 1512 g. rx x − a1x − a2x − ak; the bottom entry will
be: rak1 ak1 − a1ak1 − a2ak1 − ak
5.4 Exercises
1. a. adjA 4 −5
−1 3 ; A−1
4 7 − 5
7
− 1 7
3 7
. b. adjB 20 5
−12 −3 ; B is
not invertible.
2. a. adjA
−4 −7 −2
−10 5 −5
−1 −13 −23
; A−1
4 45
7 45
2 45
2 9 − 1
9 1 9
1 45
13 45
23 45
. b.
adjB
14 −6 −31
−7 −24 2
−35 15 −17
; B−1
− 2 27
2 63
31 189
1 27
8 63
− 2 189
5 27
− 5 63
17 189
.
3. a. adjA
183 85 74 −62
−534 −338 −379 −44
−63 −63 42 −63
−339 −388 −362 53
A−1
− 61 343
− 85 1029
− 74 1029
62 1029
178 343
338 1029
379 1029
44 1029
3 49
3 49
− 2 49
3 49
113 343
388 1029
362 1029
− 53 1029
4. a. ⟨x, y − 31 13
,− 29 13
b. ⟨x, y 2 59
,− 92 59
5. a. doesn’t apply b. ⟨x, y 3 73
,− 52 73
6. a. ⟨x, y, z ⟨ 3 4
, 7 4
, 1 2 b. doesn’t apply
7. a. ⟨x, y, z 209 193
, 66 193
, 367 193
b. ⟨x, y, z − 137 83
, 26 83
,− 15 83
8. a. ⟨x, y, z, w 164 107
,− 979 107
, 399 107
, 1029 107
b. ⟨x, y, z, w ⟨5,−8, 6, 0 9. a. ⟨x, y, z, w 161
44 , 433
88 ,− 247
176 ,− 211
44 b. ⟨x, y, z, w ⟨1,− 1
2 ,− 1
2 , 1
Selected Answers to the Exercises 61
10. b 3897 6445
; d − 836 6445
11. ⟨5, 0,−2, 7
12. −5 0
0 1 ;
0 −1
−1 0 ;
4 0 0
0 1 0
0 0 4
;
0 0 −1
0 −1 0
−1 0 0
;
1 0 3
0 1 0
0 0 1
16. b. adjA
−21 −35 27
0 14 −12
0 0 −6
. It is also upper triangular.
5.5 Exercises 1. a. WSx 16 cos x cos 3x sin 2x − 9 cos x cos 2x sin 3x − 5 sin x cos 2x cos 3x; b.
WS/4 −8, so S is linearly independent. 2. a. WSx −e2x b. WS0 −1, so S is linearly independent. 3. a. WSx −ne2kx b. WSx −n ≠ 0, so S is linearly independent. 4. a. WSx zx b. S is linearly dependent. 5. a. WSx zx b. S is linearly dependent. 6. a. WSx 18 cos2x cos22x 18 cos2x sin22x 18 sin2x cos22x 18 sin2x sin22x
b. WS0 18, so S is linearly independent. 7. a. WSx 12tan xsec22xsec23x3tan 3x − 2tan 2x
6 tan 2x sec23xsec2xtan x − 3tan 3x 4 tan 3x sec22xsec2x2tan 2x − tan x
b. WS/3 216, so S is linearly independent.
8. a. WSx 23160 x −
3 20 b. WS1 23160 , so S is linearly independent.
9. a. WSx 1144 000 x −
283 60 b. WS1 1144 000 , so S is linearly independent.
10. a. WSx − 916 1
x−1x−2x−3x−43/2 b. WS5 ≠ 0, so S is linearly independent.
11. a. WSx 5x4x3xln 4 − ln 3ln 5 − ln 3ln 5 − ln 4 b. WS0 ln 4 − ln 3ln 5 − ln 3ln 5 − ln 4 ≠ 0, so S is linearly independent.
12. a. WSx zx b. S is linearly dependent. 13. a. ek1x , ek2x ,…, eknx b. WS /x Vk1, k2,…, kn e
k1k2knx c. WS /0 Vk1, k2,…, kn ≠ 0, since the ki are distinct, so S is linearly independent.
14. a. b1 x , b2
x ,…, bnx b. WS /x Vlnb1, lnb2,…, lnbnb1 x b2
x bnx c. WS /0 Vlnb1, lnb2,…, lnbn ≠ 0, since the bi are distinct, so S is linearly independent.
15. a. xk1 , xk2 ,…, xkn b. WS /x Vk1, k2,…, knx k1k2kn−nn1/2 c.
WS /0 Vk1, k2,…, kn ≠ 0, since the ki are distinct, so S is linearly independent. 16. a. x − k1m, x − k2m, …, x − knm b. and c. If m is a positive integer and
n m 1, then WS /x zx and consequently S will be dependent, since dimm m 1, and S / contains n m 1 vectors from m. If m is not a positive integer, then m, m − 1, …, m − i is never zero for any positive integer i, and we get: WSx m mm − 1 mm − 1m − 2 m − n 2
x − k1mn−1x − k2mn−1 x − knmn−1 Vx − k1, x − k2,…, x − kn; Note: the sign or − depends on the remainder j when n is divide by 4, i.e. n 4i j, where i is a non-negative integer and j 0, 1, 2, or 3, since we will need to perform row
62 Selected Answers to the Exercises
exchanges in order to bring the Wronskian matrix into a form similar to the Vandermonde matrix (note that the powers of x − ki are in decreasing rather than increasing order); the number of these exchanges depends on j; by letting x be any number bigger than kn (where we assume the ki are in increasing order), we get a non-zero value for WS /x, so S is independent.
Selected Answers to the Exercises 63
Chapter Six Exercises
6.1 Exercises 1. p 2 − 6; EigA, 2 Span⟨−1, 1; EigA,−3 Span⟨−2, 1. Each is
1-dimensional. 2. p 2 − 8 15; EigA, 5 Span⟨2, 5; EigA, 3 Span⟨1, 2. Each is
1-dimensional. 3. p 2 − 11 − 12; EigA,−1 Span⟨−2, 3; EigA, 12 Span⟨3, 2. Each
is 1-dimensional. 4. p 2 3 − 10; EigA, 2 Span⟨−4, 3; EigA,−5 Span⟨−3, 2. Each
is 1-dimensional. 5. p 2 36; since the eigenvalues are imaginary, there are no eigenvectors. 6. p 2 − 15 44; EigA, 4 Span⟨5, 2; EigA, 11 Span⟨7, 3. Each is
1-dimensional. 7. p − 5 2 4; EigA, 5 Span⟨1, 0, 0;
EigA,−2 Span⟨4,−7, 0; EigA,−4 Span⟨2, 27, 18. Each is 1-dimensional.
8. p − 4 − 7 2; EigA, 4 Span⟨6, 2, 1; EigA, 7 Span⟨0,−3, 2; EigA,−2 Span⟨0, 0, 1. Each is 1-dimensional.
9. p 5 − 8; EigA, 0 Span⟨3,−5, 0; EigA,−5 Span⟨1, 0, 0; EigA, 8 Span⟨69,−91, 104. Each is 1-dimensional.
10. p − 32 − 2 − 4; EigA, 3 Span⟨1, 0, 0, 0,⟨0, 5, 1, 0, 2-dimensional; EigA, 2 Span⟨−3, 1, 0, 0; EigA, 4 Span⟨27,−9,−2, 2; the other two are 1-dimensional.
11. p 22 − 32; EigA,−2 Span⟨5, 4, 0, 0,⟨0, 0,−1, 3; EigA, 3 Span⟨0,−1, 2, 0,⟨0, 0, 0, 1. Each is 2-dimensional.
12. p − 53 3; EigA, 5 Span⟨0, 0, 1, 0,⟨0, 0, 0, 1,⟨8, 7, 0, 0, 3-dimensional, and EigA,−3 Span⟨0, 1,−3, 2, 1-dimensional.
13. p 2 − − 10/9; EigA, 5/3 Span⟨−7, 4; EigA,−2/3 Span⟨1,−1. Each is 1-dimensional.
14. p 1/3 − 4/3 − 2/3; EigA,−1/3 Span⟨1, 0, 0; EigA, 4/3 Span⟨1, 1, 0; EigA, 2/3 Span⟨3, 1, 2. Each is 1-dimensional.
15. p − 5/2 3/2 − 1/2; EigA, 5/2 Span⟨1, 0, 0, 0; EigA, 0 Span⟨7, 5, 0, 0; EigA,−3/2 Span⟨11, 12,−4, 0; EigA, 1/2 Span⟨57, 34, 10,−8. Each is 1-dimensional.
16. a. p 2 − 36; EigA, 6 Span⟨3, 2; EigA,−6 Span⟨−3, 2. b. the eigenvalues are imaginary: 6i, so there are no eigenvectors.
17. a. p − 32 2; EigA, 3 Span⟨1, 0, 0,⟨0, 2, 5, 2-dimensional; EigA,−2 Span⟨−3, 1, 0, 1-dimensional. b. p − 32 2; EigA, 3 Span⟨1, 0, 0;
64 Selected Answers to the Exercises
EigA,−2 Span⟨−14, 5, 0; both 1-dimensional. 18. a. p 72 − 2; EigA,−7 Span⟨3, 1, 0,⟨0, 0, 1, 2-dimensional;
EigA, 2 Span⟨0, 1,−2, 1-dimensional. b. p 72 − 2; EigA,−7 Span⟨0, 0, 1; EigA, 2 Span⟨0, 1, 2; each is 1-dimensional.
19. a. p − 32 22; EigA,−2 Span⟨1, 0, 0, 0, 1-dimensional; EigA, 3 Span⟨−2, 1, 0, 0,⟨49, 0, 15, 5, 2-dimensional. b. p − 32 22; EigA,−2 Span⟨1, 0, 0, 0,⟨0, 7, 5, 0; EigA, 3 Span⟨−2, 1, 0, 0,⟨46, 0, 15, 5. Both are 2-dimensional. c. p − 32 22; EigA,−2 Span⟨1, 0, 0, 0,⟨0, 7, 5, 0, 2-dimensional; EigA, 3 Span⟨−2, 1, 0, 0, 1-dimensional.
20. a. p − 3 23; EigA,−2 Span⟨1, 0, 0, 0; EigA, 3 Span⟨−2, 1, 0, 0. Both are 1-dimensional. b. p − 3 23; EigA, 3 Span⟨−4, 1, 0, 0, 1-dimensional; EigA,−2 Span⟨1, 0, 0, 0,⟨0, 2, 5, 0, 2-dimensional. c. p − 3 23; EigA, 3 Span⟨−4, 1, 0, 0, 1-dimensional; EigA,−2 Span⟨1, 0, 0, 0,⟨0, 2, 5, 0,⟨0,−4, 0, 5, 3-dimensional.
21. a. p − 32 − 13; EigA, 1 Span⟨1, 0, 0, 0, 0,⟨0, 3, 1, 0, 0; EigA, 3 Span⟨2, 1, 0, 0, 0,⟨0, 0, 3, 1, 0. Both are 2-dimensional. b. p − 32 − 13; EigA, 1 Span⟨1, 0, 0, 0, 0,⟨0, 3, 1, 0, 0,⟨0, 0, 0,−5, 2, 3-dimensional; EigA, 3 Span⟨2, 1, 0, 0, 0,⟨0, 0, 3, 1, 0, 2-dimensional. c. p − 32 − 13; EigA, 1 Span⟨1, 0, 0, 0, 0,⟨0, 3, 1, 0, 0,⟨0, 5, 0,−5, 2, 3-dimensional; EigA, 3 Span⟨2, 1, 0, 0, 0, 1-dimensional.
22. a. p − 3 2 − 2 ; Eig A, 3 Span⟨1, 1, 0,⟨0, 0, 1,
2-dimensional; Eig A, 2 Span 0, 3 − 2 , 5 , 1-dimensional. b. p − 3
2 − 2 ; Eig A, 3 Span⟨0, 0, 1, 1-dimensional;
Eig A, 2 Span 0, 3 − 2 , 5 , 1-dimensional. 23. a. p − 322 − 2; EigA, 32 Span⟨0, 0, 1, 1-dimensional;
EigA, 2 Span⟨0, 3 − 2, 1, 1-dimensional. b. p − 322 − 2; EigA, 32 Span⟨, 2, 0,⟨0, 0, 1, 2-dimensional; EigA, 2 Span⟨0, 3 − 2, 1, 1-dimensional.
24. A −8 5
−10 7 . We get the same characteristic polynomials and thus same
eigenvalues. However, for A, EigA,−3 Span⟨1, 1 and EigA, 2 Span⟨1, 2. These eigenspaces are different from the eigenspaces for A. Notice, however, that the corresponding eigenspaces are orthogonal to each other!
31. a. 2 − 2 cos 1; b. The discriminant is −4 sin2, which is negative unless sin 0, which corresponds to n. In this case, cosn 1, and R I.
32. a. D −1 0
0 1 b. rotate a vector v counterclockwise by then reflect this resulting
Selected Answers to the Exercises 65
vector across the y-axis; c. 2 − 1 d. the eigenvalues are always 1 and −1; e. EigA,−1 Span⟨sin, 1 cos and EigA, 1 Span⟨sin,−1 cos. f. EigA,−1 Span⟨sin/2, cos/2 and EigA, 1 Span⟨cos/2,−sin/2. h. they are orthogonal to each other!
i. 5/13 12/13
12/13 −5/13 ; EigA,−1 Span⟨−2, 3 and EigA, 1 Span⟨3, 2.
j. Repeat (a) to (h) for the matrix B:
a. D 1 0
0 −1 b. reflect v across the x-axis, then rotate this resulting vector
counterclockwise by . c. 2 − 1 d. the eigenvalues are always 1 and −1; e. EigA,−1 Span⟨sin,−1 − cos and EigA, 1 Span⟨sin, 1 − cos. f. EigA,−1 Span⟨sin/2,−cos/2 and EigA, 1 Span⟨cos/2, sin/2. h. again, they are orthogonal to each other.
i. −5/13 12/13
12/13 5/13 ; EigA,−1 Span⟨−3, 2 and EigA, 1 Span⟨2, 3.
34. b. EigA1 ⊕ A2,−5 Span⟨0, 0, 1, 2, 1; EigA1 ⊕ A2, 3 Span⟨−2, 1, 0, 0, 0,⟨0, 0,−1, 1, 0, ⟨0, 0, 1, 0, 1; EigA1 ⊕ A2, 7 Span⟨−5, 2, 0, 0, 0
6.2 Exercises 1. For −5 : ⟨1, 1, 0; for 3 : ⟨1, 1, 1 and for 7 : ⟨0,−1, 1. 2. Hint: the exponent of p1 can be 0, 1, 2, …, n1. 3. 24 possibilities: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; roots are: 5, 6,−3 4. 24 possibilities: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; roots are: 6,−3,−4 5. 8 possibilities: 1, 3, 5, 15; roots are: 5, 3 6 , 3 − 6 . 6. p 3 − 92 23 − 15 − 5 − 1 − 3; for 1 : ⟨−1, 0, 1, dim 1;
for 3 : ⟨1, 0, 1, dim 1; for 5 : ⟨0, 1, 0, dim 1. 7. p 3 − 22 − 15 36 − 32 4;
for 3 : ⟨0, 0, 1, dim 1; for −4 : ⟨−7, 0, 1, dim 1. 8. p 3 − 152 72 − 112 − 7 − 42;
for 7 : ⟨1, 1, 1, dim 1; for 4 : ⟨−1, 1, 0,⟨−1, 0, 1, dim 2. 9. p 3 − 52 − 7 35; for 5 : ⟨0, 0, 1, dim 1;
for 7 : 1, 7 − 3, 0 , dim 1; for − 7 : 1,− 7 − 3, 0 , dim 1. 10. p 3 − 32 − 10 24 − 2 − 4 3; for −3 : ⟨2, 9, 2, dim 1;
for 4 : ⟨1, 1, 1, dim 1; for 2 : ⟨36, 42, 31, dim 1. 11. p 3 − 7/42 7/16 15/64; for −1/4 : ⟨2, 3, 2, dim 1; for
3/4 : ⟨1, 1, 1, dim 1; for 5/4 : ⟨4, 4, 3, dim 1.
66 Selected Answers to the Exercises
12. p 3 − 13 − 12; for 4 : ⟨0,−1, 1, dim 1; for −1 : ⟨−1, 0, 1, dim 1; for −3 : ⟨2,−3, 0, dim 1.
13. p 3 − 152 72 − 112; for 4 : ⟨4, 0, 5,⟨2,−5, 0, dim 2; for 7 : ⟨1,−1, 1, dim 1.
14. p 3 − 152 72 − 112 (note: same as Exercise 13); for 4 : ⟨2,−1, 1, dim 1; for 7 : ⟨1,−1, 1, dim 1.
15. p 3 2 − 21 − 45; for 5 : ⟨−4, 2, 1, dim 1; for −3 : ⟨−2, 1, 0,⟨1, 0, 1, dim 2;
16. p 3 − 52 − 32 − 36; for 9 : ⟨1, 4,−2, dim 1; for −2 : ⟨1, 1, 0,⟨1, 0, 1, dim 2;
17. p 3 − 72 − 5 75; for −3 : ⟨1, 3,−2, dim 1; for 5 : ⟨0, 1,−1,⟨2, 3, 0, dim 2;
18. p 3 1 3 2 − 40
9 − 112
27 ; for 7/3 : ⟨1, 1, 2, dim 1; for
−4/3 : ⟨−2, 5, 0,⟨3, 0, 5, dim 2; 19. p 3 1
4 2 − 33
16 63
64 ; for −7/4 : ⟨−2,−1, 2, dim 1; for
3/4 : ⟨3, 1, 0,⟨3, 0, 5, dim 2; 20. p 3 − 2
5 2 − 3
5 36
125 ; for −4/5 : ⟨−1,−1, 2, dim 1; for
3/5 : ⟨2, 1, 0,⟨3, 0, 5, dim 2; 21. p 4 − 252 − 33 75; for −5 : ⟨3, 0,−5, 4, dim 1; for
0 : ⟨−4, 0, 0, 3, dim 1; for 3 : ⟨0, 1, 0, 0, dim 1; for 5 : ⟨3, 0, 5, 4, dim 1.
22. p 4 − 982 2401 − 72 72; for −7 : ⟨−1, 0, 0, 1,⟨0,−1, 1, 0, dim 2; for 7 : ⟨1, 0, 0, 1,⟨0, 1, 1, 0, dim 2.
23. p 4 − 1162 1600 10 − 4 4 − 10; for −10 : ⟨1,−1, 1,−1, dim 1; for 10 : ⟨1, 1, 1, 1, dim 1; for −4 : ⟨−1,−1, 1, 1, dim 1; for 4 : ⟨−1, 1, 1,−1, dim 1.
24. p 4 − 73 2 63 − 90 − 2 − 3 3 − 5; for −3 : ⟨0,−1, 0, 1, dim 1; for 3 : ⟨3, 0, 1, 0, dim 1; for 2 : ⟨−9, 1,−3,−2, dim 1; for 5 : ⟨−1,−1, 0, 1, dim 1.
25. p 4 − 33 − 122 20 48 − 3 − 4 22; for 3 : ⟨2,−1, 1,−1, dim 1; for 4 : ⟨−5, 2,−2, 0, dim 1; for −2 : ⟨3, 0, 3,−1,⟨−3, 3, 0, 1, dim 2.
26. p 4 23 − 232 − 24 144 − 32 42; for 3 : ⟨1, 0, 2, 0,⟨0, 1,−2, 1, dim 2; for −4 : ⟨−2, 1,−5, 1, dim 1.
27. p 4 − 3 − 182 52 − 40 − 23 5; for 2 : ⟨1, 0, 2, 0,⟨0, 1, 3, 0,⟨0, 0,−3, 1, dim 3; for −5 : ⟨−2, 1,−3, 1, dim 1.
28. p 4 − 53 62 4 − 8 − 23 1; for 2 : ⟨−2, 5, 1, 0,⟨−5, 10, 0, 4, dim 2; for −1 : ⟨−2, 1,−3, 7, dim 1.
29. p 5 − 104 323 − 322; for 2 : ⟨0, 0, 1, 0, 0, dim 1. for 0 : ⟨0,−1, 0, 1, 0,⟨1, 0, 0, 0,−1, dim 2; for 4 : ⟨0, 1, 0, 1, 0,⟨1, 0, 0, 0, 1, dim 2.
30. p 3 2 − 31 46; EigA,−6. 6758 Span⟨−0. 60015,−0. 8689, 1; EigA, 1. 7594 Span⟨1. 10664, 0. 3865, 1; EigA, 3. 9164 Span⟨−1. 72078, 2. 3395, 1.
31. p 3 82 7 − 13; EigA,−6. 6545 Span⟨−0. 5515, 0. 1185, 1;
Selected Answers to the Exercises 67
EigA,−2. 2239 Span⟨0. 92536,−4. 1326, 1; EigA, 0. 878 43 Span⟨1. 9595, 0. 680745, 1.
32. p 4 − 33 − 142 26 10; EigA,−3. 3149 Span⟨−0. 158 774, 0. 156133,−0. 07236 9, 1; EigA,−0. 33044 Span⟨−2. 75815,−5. 4277, 8. 15926, 1; EigA, 1. 9403 Span⟨6. 3174, 1. 3771, 2. 929, 1; EigA, 4. 705 Span⟨2. 3495,−5. 35553,−2. 89094, 1.
37. a. False. b. True. c. False. d. False. e. True. f. False. g. True.
6.3 Exercises Note: the diagonal entries of D can be rearranged, as long as the corresponding eigenvectors are also located in the corresponding columns of C.
1. D −3 0
0 2 ; C
−2 −1
1 1 2. D
−5 0
0 2 ; C
−3 −4
2 3
3. This matrix is not diagonalizable because the eigenvalues are imaginary.
4. D −2/3 0
0 5/3 ; C
1 −7
−1 4 5. D
−4 0 0
0 −2 0
0 0 5
;
C
2 4 1
27 −7 0
18 0 0
6. D
−1/3 0 0
0 2/3 0
0 0 4/3
; C
1 3 1
0 1 1
0 2 0
7. D
2 0 0 0
0 3 0 0
0 0 3 0
0 0 0 4
;
C
−3 1 0 27
1 0 5 −9
0 0 1 −2
0 0 0 2
8. D
−5 0 0
0 3 0
0 0 7
; C
1 1 0
1 1 −1
0 1 1
9. D
1 0 0
0 3 0
0 0 5
;
C
−1 1 0
0 0 1
1 1 0
10. This matrix is not diagonalizable because there are only two linearly independent vectors, and this is a 3 3 matrix.
68 Selected Answers to the Exercises
11. D
4 0 0
0 4 0
0 0 7
; C
−1 −1 1
1 0 1
0 1 1
12. D
−3 0 0
0 2 0
0 0 4
;
C
2 36 1
9 42 1
2 31 1
13. D
−1/4 0 0
0 3/4 0
0 0 5/4
; C
2 1 4
3 1 4
2 1 3
14. D
−3 0 0
0 −1 0
0 0 4
;
C
2 −1 0
−3 0 −1
0 1 1
15. D
4 0 0
0 4 0
0 0 7
; C
4 2 1
0 −5 −1
5 0 1
16. This matrix is not diagonalizable because there are only two linearly independent vectors, and this is a 3 3 matrix.
17. D
−3 0 0
0 −3 0
0 0 5
; C
−2 1 −4
1 0 2
0 1 1
18. D
−2 0 0
0 −2 0
0 0 9
;
C
1 0 1
0 1 4
1 −1 −2
19. D
−7 0 0 0
0 −7 0 0
0 0 7 0
0 0 0 7
; C
0 −1 0 1
−1 0 1 0
1 0 1 0
0 1 0 1
20. D
−3 0 0 0
0 2 0 0
0 0 3 0
0 0 0 5
;
C
0 −9 3 −1
−1 1 0 −1
0 −3 1 0
1 −2 0 1
Selected Answers to the Exercises 69
21. D
−2 0 0 0
0 −2 0 0
0 0 3 0
0 0 0 4
; C
3 −3 2 −5
0 3 −1 2
3 0 1 −2
−1 1 −1 0
22. This matrix is not diagonalizable because there are only three linearly independent vectors, and this is a 4 4 matrix.
23. D
−5 0 0 0
0 2 0 0
0 0 2 0
0 0 0 2
; C
−2 1 0 0
1 0 1 0
−3 2 3 −3
1 0 0 1
24. D
0 0 0 0 0
0 0 0 0 0
0 0 2 0 0
0 0 0 4 0
0 0 0 0 4
;
C
1 0 0 0 1
0 −1 0 1 0
0 0 1 0 0
0 1 0 1 0
−1 0 0 0 1 25. Only the matrix in (a) is diagonalizable. 26. Only the matrix in (a) is diagonalizable. 27. Only the matrix in (a) is diagonalizable. 28. Only the matrix in (b) is diagonalizable. 29. Only the matrix in (c) is diagonalizable. 30. Only the matrix in (b) is diagonalizable. 31. Only the matrix in (a) is diagonalizable. 32. Only the matrix in (b) is diagonalizable.
33. −28381 −37884
18942 25288 34.
22003 729
22099 729
− 12628 729
− 12724 729
35.
3125 1804 −3167
0 −32 −1488
0 0 −1024
36.
243 −1477 1261 14472
0 32 1804 −2996
0 0 −3125 −461
0 0 0 1024
37.
−6493 3368 3368
−23300 20175 3368
16564 −16564 243
38.
−5322 −362 6708
−4749 −605 6378
−5354 −362 6740
70 Selected Answers to the Exercises
39.
483 484 484
−3801 −2777 −3801
3075 2050 3074
40.
−77891 −31566 63132
78915 32590 −63132
−78915 −31566 64156
41.
−59113 59081 59081
−236324 236292 236324
118162 −118162 −118194
42.
0 0 0 16807
0 0 16807 0
0 16807 0 0
16807 0 0 0
43.
3125 −1899 −8646 −1899
3368 −518 −10104 −275
0 −633 243 −633
−3368 550 10104 307
44.
7448 8030 −8030 −1650
−3212 −3519 3487 825
3212 3487 −3519 −825
−1100 −1375 1375 793
45.
−12596 −18942 6314 18942
6314 9503 −3157 −9471
−18942 −28413 9503 28413
6314 9471 −3157 −9439
46.
512 0 0 0 512
0 512 0 512 0
0 0 32 0 0
0 512 0 512 0
512 0 0 0 512
47. 0 11664
5184 0 48.
243 825 −330
0 −32 110
0 0 243
49.
−16807 0 0
−5613 32 0
11226 −33678 −16807
50.
−32 −550 770 220
0 243 −385 1155
0 0 −32 825
0 0 0 243
51.
−32 −1100 440 −880
0 243 −110 220
0 0 −32 0
0 0 0 −32
52.
1 484 −1452 4356 10890
0 243 −726 2178 5445
0 0 1 726 1815
0 0 0 243 605
0 0 0 0 1
61. a. False b. False c. False d. True e. False f. True g. False h. True i. True j. False.
6.4 Exercises We provide the answers for etA. To get eA, just replace t with 1.
Selected Answers to the Exercises 71
1. −8e2t 9e−5t −12e2t 12e−5t
6e2t − 6e−5t 9e2t − 8e−5t
2. − 4
3 e−
2 3
t 7 3
e 5 3
t − 7 3
e− 2 3
t 7 3
e 5 3
t
4 3
e− 2 3
t − 4 3
e 5 3
t 7 3
e− 2 3
t − 4 3
e 5 3
t
3.
e5t − 4 7
e−2t 4 7
e5t 6 7
e−2t 1 9
e−4t − 61 63
e5t
0 e−2t − 3 2
e−2t 3 2
e−4t
0 0 e−4t
4.
e3t −3e2t 3e3t 15e2t − 15e3t 3 2
e2t − 15e3t 27 2
e4t
0 e2t −5e2t 5e3t − 1 2
e2t 5e3t − 9 2
e4t
0 0 e3t e3t − e4t
0 0 0 e4t
5.
−e3t 2e−5t e3t − e−5t e3t − e−5t
−e3t 2e−5t − e7t e3t − e−5t e7t e3t − e−5t
−e3t e7t e3t − e7t e3t
6.
36 5
e2t − 22 35
e−3t − 39 7
e4t 2 7
e−3t − 2 7
e4t − 36 5
e2t 12 35
e−3t 48 7
e4t
42 5
e2t − 99 35
e−3t − 39 7
e4t 9 7
e−3t − 2 7
e4t − 42 5
e2t 54 35
e−3t 48 7
e4t
31 5
e2t − 22 35
e−3t − 39 7
e4t 2 7
e−3t − 2 7
e4t − 31 5
e2t 12 35
e−3t 48 7
e4t
7.
3e−t − 2e−3t 2e−t − 2e−3t 2e−t − 2e−3t
3e−3t − 3e4t 3e−3t − 2e4t 3e−3t − 3e4t
−3e−t 3e4t −2e−t 2e4t −2e−t 3e4t
8.
6e4t − 5e7t 2e4t − 2e7t −4e4t 4e7t
−5e4t 5e7t −e4t 2e7t 4e4t − 4e7t
5e4t − 5e7t 2e4t − 2e7t −3e4t 4e7t
9.
2e−2t − e9t −e−2t e9t −e−2t e9t
4e−2t − 4e9t −3e−2t 4e9t −4e−2t 4e9t
−2e−2t 2e9t 2e−2t − 2e9t 3e−2t − 2e9t
72 Selected Answers to the Exercises
10.
1 2
e−7t 1 2
e7t 0 0 − 1 2
e−7t 1 2
e7t
0 1 2
e−7t 1 2
e7t − 1 2
e−7t 1 2
e7t 0
0 − 1 2
e−7t 1 2
e7t 1 2
e−7t 1 2
e7t 0
− 1 2
e−7t 1 2
e7t 0 0 1 2
e−7t 1 2
e7t
11.
e5t 9e2t − 9e3t 3e3t − 3e5t 9e2t − 9e3t
−e−3t e5t −e2t 2e−3t 3e−3t − 3e5t −e2t e−3t
0 3e2t − 3e3t e3t 3e2t − 3e3t
e−3t − e5t 2e2t − 2e−3t −3e−3t 3e5t 2e2t − e−3t
12.
−12e−2t 8e3t 5e4t −15e−2t 10e3t 5e4t 15e−2t − 10e3t − 5e4t 6e−2t − 6e3t
6e−2t − 4e3t − 2e4t 8e−2t − 5e3t − 2e4t −7e−2t 5e3t 2e4t −3e−2t 3e3t
−6e−2t 4e3t 2e4t −7e−2t 5e3t 2e4t 8e−2t − 5e3t − 2e4t 3e−2t − 3e3t
4e−2t − 4e3t 5e−2t − 5e3t −5e−2t 5e3t −2e−2t 3e3t
13.
−3e2t 4e−5t −6e2t 6e−5t 2e2t − 2e−5t 6e2t − 6e−5t
2e2t − 2e−5t 4e2t − 3e−5t −e2t e−5t −3e2t 3e−5t
−6e2t 6e−5t −9e2t 9e−5t 4e2t − 3e−5t 9e2t − 9e−5t
2e2t − 2e−5t 3e2t − 3e−5t −e2t e−5t −2e2t 3e−5t
14.
1 2
e4t 1 2
0 0 0 1 2
e4t − 1 2
0 1 2
e4t 1 2
0 1 2
e4t − 1 2
0
0 0 e2t 0 0
0 1 2
e4t − 1 2
0 1 2
e4t 1 2
0 1 2
e4t − 1 2
0 0 0 1 2
e4t 1 2
15. 1 2
e−6t 1 2
e6t − 3 4
e−6t 3 4
e6t
− 1 3
e−6t 1 3
e6t 1 2
e−6t 1 2
e6t
16.
e3t −3e−2t 3e3t 6 5
e−2t − 6 5
e3t
0 e−2t − 2 5
e−2t 2 5
e3t
0 0 e3t
17.
e−7t 0 0
− 1 3
e2t 1 3
e−7t e2t 0 2 3
e2t − 2 3
e−7t −2e2t 2e−7t e−7t
Selected Answers to the Exercises 73
18.
e−2t 2e−2t − 2e3t − 14 5
e−2t 14 5
e3t − 4 5
e−2t 4 5
e3t
0 e3t 7 5
e−2t − 7 5
e3t − 21 5
e−2t 21 5
e3t
0 0 e−2t −3e−2t 3e3t
0 0 0 e3t
19.
e−2t 4e−2t − 4e3t − 8 5
e−2t 8 5
e3t 16 5
e−2t − 16 5
e3t
0 e3t 2 5
e−2t − 2 5
e3t − 4 5
e−2t 4 5
e3t
0 0 e−2t 0
0 0 0 e−2t
20.
et −2et 2e3t 6et − 6e3t −18et 18e3t −45et 45e3t
0 e3t 3et − 3e3t −9et 9e3t − 45 2
et 45 2
e3t
0 0 et −3et 3e3t − 15 2
et 15 2
e3t
0 0 0 e3t − 5 2
et 5 2
e3t
0 0 0 0 et
6.5 Exercises 1. a. ⟨vB ⟨−3, 7,−10 and ⟨vB/ ⟨3, 8/3, 4/3. b. The rrefs contained I3 on the left side.
c. CB,B/
−2 1 1 4 3
0 − 2 3
− 1 3
1 2 3
2. a. ⟨vB ⟨4,−3, 6, 33/2 and ⟨vB/ ⟨5, 3,−7/2, 15/2. b. The rrefs contained I4 on the left side.
c. CB,B/
1 2 −3 2
0 1 1 0
−1 −1 1 −1
0 2 −6 3
.
3. a. Tv − 49 2 ⟨0,−1, 1 15⟨1,−1, 1 − 23⟨1, 2, 1 ⟨−8,−73/2,−65/2
b. T
−1 4 −3 3
− 5 2 − 7
2 13 2
5
− 1 2
7 2
− 1 2
10
74 Selected Answers to the Exercises
4. a.
4 3 1
−3 1 0
−5 −2 4
0 −1 −2
−3
7
−10
−1
16
−39
13
. Decoding:
Tv −1⟨1, 0, 1, 2 16⟨0, 1, 1,−1 − 39⟨0, 0, 2, 1 13⟨0, 0, 0,−1 ⟨−1, 16,−63,−70.
b. T
1 0 0 0
0 1 0 0
1 1 2 0
2 −1 1 −1
4 3 1
−3 1 0
−5 −2 4
0 −1 −2
1 1 0
0 1 1
−1 2 1
−1
3 2 −1
−1 −2 2
−9 9 0
1 13 −5
5. a.
6 −3 −1
−2 1 0
−7 2 4
−3
7
−10
−29
13
−5
. Decoding, we get:
Tv −29⟨1, 0,−1 13⟨1, 1, 2 − 5⟨0, 1, 1 ⟨−16, 8, 50.
b. T
1 1 0
0 1 1
−1 2 1
6 −3 −1
−2 1 0
−7 2 4
1 1 0
0 1 1
−1 2 1
−1
3 2
3 2
− 5 2
−5 0 4
− 15 2
− 9 2
19 2
6. a.
7 3 1
−1 −4 0
3 5 −2
4
−3
7
26
8
−17
b.
0 3 2
− 3 2
6 21 2
11 2
−7 − 31 2
− 19 2
7. a. B v1, v2, v4 ⟨−3, 1, 6,−5,⟨4, 2,−4,−4,⟨1, 4, 7, 3 b. B/ ⟨1, 0, 0, 7,⟨0, 1, 0,−8,⟨0, 0, 1, 4 c. ⟨18, 4,−24,−2 −2⟨−3, 1, 6,−5 3⟨4, 2,−4,−4 d. ⟨18, 4,−24,−2 18⟨1, 0, 0, 7 4⟨0, 1, 0,−8 − 24⟨0, 0, 1, 4
e. CB,B/
−3 4 18
1 2 4
6 −4 −24
f.
−3 4 18
1 2 4
6 −4 −24
−2
3
0
18
4
−24 8. a. B v1, v2, v3 ⟨−3, 1, 6,−5,⟨4, 2,−4,−4,⟨1, 4, 7, 3
b. B/ ⟨1, 0, 0, 7,⟨0, 1, 0,−8,⟨0, 0, 1, 4 c. ⟨−10,−3, 1,−42 5⟨−3, 1, 6,−5 2⟨4, 2,−4,−4 − 3⟨1, 4, 7, 3 d. ⟨−10,−3, 1,−42 −10⟨1, 0, 0, 7 − 3⟨0, 1, 0,−8 1⟨0, 0, 1, 4
Selected Answers to the Exercises 75
e. CB,B/
−3 4 1
1 2 4
6 −4 7
f.
−3 4 1
1 2 4
6 −4 7
5
2
−3
−10
−3
1
9. a. B v1, v2, v3 ⟨−3, 12, 5, 2,−2,⟨1,−4, 4, 3,−4,⟨4,−16,−6,−4, 18 b. B/ ⟨1,−4, 0, 0, 3,⟨0, 0, 1, 0, 5,⟨0, 0, 0, 1,−9 For (c) and (d), there are no vectors from S which are not in B.
e. CB,B/
−3 1 4
5 4 −6
2 3 −4 10. a. B v1, v2, v3 ⟨−3,−4,−2, 9, 1, 1,⟨1, 2, 4, 9, 11,−11,⟨4, 3, 5, 16, 1, 8
b. B/ ⟨1, 0, 0, 3,−5, 9,⟨0, 1, 0,−7, 2,−6,⟨0, 0, 1, 5, 3,−2 c. ⟨−21,−36,−26, 59,−45, 79 8⟨−3,−4,−2, 9, 1, 1 − 5⟨1, 2, 4, 9, 11,−11 2⟨4, 3, 5, 16, 1, 8 ⟨−20,−37,−23, 84,−43, 88 9⟨−3,−4,−2, 9, 1, 1 − 5⟨1, 2, 4, 9, 11,−11 3⟨4, 3, 5, 16, 1, 8 d. ⟨−21,−36,−26, 59,−45, 79 −21⟨1, 0, 0, 3,−5, 9 − 36⟨0, 1, 0,−7, 2,−6 − 26⟨0, 0, 1, 5, 3,−2 ⟨−20,−37,−23, 84,−43, 88 −20⟨1, 0, 0, 3,−5, 9 − 37⟨0, 1, 0,−7, 2,−6 − 23⟨0, 0, 1, 5, 3,−2
e. CB,B/
−3 1 4
−4 2 3
−2 4 5
f.
−3 1 4
−4 2 3
−2 4 5
8
−5
2
−21
−36
−26
−3 1 4
−4 2 3
−2 4 5
9
−5
3
−20
−37
−23 11. a. B v1, v2, v4, v5
⟨−5, 3,−3, 2,−14,−4,⟨3,−4,−7,−5,−21, 7,⟨2,−1, 2, 0, 11, 2,⟨−1, 2, 5, 3, 17,−8 b. B/ ⟨1, 0, 3, 0, 7, 0,⟨0, 1, 4, 0, 3, 0,⟨0, 0, 0, 1, 6, 0,⟨0, 0, 0, 0, 0, 1 c. ⟨−21, 17, 5, 16, 0,−26 3⟨−5, 3,−3, 2,−14,−4 − 2⟨3,−4,−7,−5,−21, 7 d. ⟨−21, 17, 5, 16, 0,−26 −21⟨1, 0, 3, 0, 7, 0 17⟨0, 1, 4, 0, 3, 0
16⟨0, 0, 0, 1, 6, 0 − 26⟨0, 0, 0, 0, 0, 1
76 Selected Answers to the Exercises
e. CB,B/
−5 3 2 −1
3 −4 −1 2
2 −5 0 3
−4 7 2 −8
f.
−5 3 2 −1
3 −4 −1 2
2 −5 0 3
−4 7 2 −8
3
−2
0
0
−21
17
16
−26 12. a. B v1, v2, v4, v5
⟨−4,−5,−1, 3, 7, 1,⟨2, 3,−1,−1,−8, 9,⟨−1, 0,−4, 2, 2, 1,⟨3, 2, 6,−5,−4,−12 b. B/ ⟨1, 0, 4, 0, 0, 9,⟨0, 1,−3, 0, 0,−6,⟨0, 0, 0, 1, 0, 7,⟨0, 0, 0, 0, 1,−2 c. ⟨−2,−1,−5, 3,−10, 29 2⟨−4,−5,−1, 3, 7, 1 3⟨2, 3,−1,−1,−8, 9 d. ⟨−2,−1,−5, 3,−10, 29 −2⟨1, 0, 4, 0, 0, 9 − ⟨0, 1,−3, 0, 0,−6
3⟨0, 0, 0, 1, 0, 7 − 10⟨0, 0, 0, 0, 1,−2
e. CB,B/
−4 2 −1 3
−5 3 0 2
3 −1 2 −5
7 −8 2 −4
f.
−4 2 −1 3
−5 3 0 2
3 −1 2 −5
7 −8 2 −4
2
3
0
0
−2
−1
3
−10 13. a. B v1, v2, v3 ⟨−3, 1, 4,−21,−20,⟨−4, 2, 3,−36,−37,⟨−2, 4, 5,−26,−23
b. B/ ⟨1, 0, 0, 8, 9,⟨0, 1, 0,−5,−5,⟨0, 0, 1, 2, 3 c. ⟨9, 9, 16, 59, 84 3⟨−3, 1, 4,−21,−20 − 7⟨−4, 2, 3,−36,−37 5⟨−2, 4, 5,−26,−23 ⟨1, 11, 1,−45,−43 −5⟨−3, 1, 4,−21,−20 2⟨−4, 2, 3,−36,−37 3⟨−2, 4, 5,−26,−23 ⟨1,−11, 8, 79, 88 9⟨−3, 1, 4,−21,−20 − 6⟨−4, 2, 3,−36,−37 − 2⟨−2, 4, 5,−26,−23 d. ⟨9, 9, 16, 59, 84 9⟨1, 0, 0, 8, 9 9⟨0, 1, 0,−5,−5 16⟨0, 0, 1, 2, 3 ⟨1, 11, 1,−45,−43 ⟨1, 0, 0, 8, 9 11⟨0, 1, 0,−5,−5 ⟨0, 0, 1, 2, 3 ⟨1,−11, 8, 79, 88 ⟨1, 0, 0, 8, 9 − 11⟨0, 1, 0,−5,−5 8⟨0, 0, 1, 2, 3
e. CB,B/
−3 −4 −2
1 2 4
4 3 5
f.
−3 −4 −2
1 2 4
4 3 5
3
−7
5
9
9
16
Selected Answers to the Exercises 77
−3 −4 −2
1 2 4
4 3 5
−5
2
3
1
11
1
−3 −4 −2
1 2 4
4 3 5
9
−6
−2
1
−11
8
14. a. B v1, v2, v3, v4 ⟨−4,−5, 3, 19, 2,−8,⟨−8,−1, 2,−28, 3,−26,⟨2, 2,−1,−5, 0, 15,⟨7, 3,−4, 5,−5,−8 b. B/ ⟨1, 0, 0, 5, 0, 8,⟨0, 1, 0,−6, 0, 3,⟨0, 0, 1, 3, 0, 7,⟨0, 0, 0, 0, 1, 9 c. ⟨8, 7,−10,−32,−8,−57 5⟨−4,−5, 3, 19, 2,−8 4⟨−8,−1, 2,−28, 3,−26
9⟨2, 2,−1,−5, 0, 15 6⟨7, 3,−4, 5,−5,−8 d. ⟨8, 7,−10,−32,−8,−57 8⟨1, 0, 0, 5, 0, 8 7⟨0, 1, 0,−6, 0, 3
− 10⟨0, 0, 1, 3, 0, 7 − 8⟨0, 0, 0, 0, 1, 9
e. CB,B/
−4 −8 2 7
−5 −1 2 3
3 2 −1 −4
2 3 0 −5
f.
−4 −8 2 7
−5 −1 2 3
3 2 −1 −4
2 3 0 −5
5
4
9
6
8
7
−10
−8
6.6 Exercises
1. a. ⟨vB ⟨−11, 6, 9 and ⟨vB/ ⟨3, 0, 2. c. CB,B/
−1 − 4 3
0
1 1 3
1
0 1 3
0
.
2. a. ⟨vB ⟨2, 4, 1,−5 and ⟨vB/ ⟨5,−3, 16,−70. c. CB,B/
1 2 5 2
1 −1 −1 0
4 6 14 6
−19 −27 −64 −28
.
3. a. Tv 27x − x2 − 491 x 382 − x2 27 − 22x − 65x2
b. TS,S/
2 17 26 18 7 2
29 2
19 2
17 3 2
11 2 − 15
2 13
.
78 Selected Answers to the Exercises
4. a. Tv 27 36x − 169x2 144x3. b. TS,S/
2 3 −2
3 −2 −6
−14 −6 19
10 8 −16
5. a. Tv −21 29x − 29x2. b. BS 1 1 1
0 −1 1
1 1 0
and BS −1
−1 1 2
1 −1 −1
1 0 −1
.
c. TS
−3 1 4
2 2 −3
−3 2 3
. e. detT −7. f. Yes. T−1 B
5 7
1 7 − 2
7
− 3 7 − 2
7 − 3
7
− 1 7 − 3
7 − 1
7
.
6. a. Tv 116 − 63x 27x2 19x3.
b. BS
1 2 5 2
1 −1 −1 0
0 1 0 0
−1 0 0 0
and BS −1
0 0 0 −1
0 0 1 0
0 −1 −1 −1 1 2
5 2
3 2
3
.
c. TS
− 13 2 − 99
2 − 51
2 −51
9 2
57 2
33 2
33
−2 −11 −5 −12
− 3 2 − 19
2 − 13
2 −12
d. detT 0. e. No.
7. a. DB
2 3 −1
−2 −3 1
0 −1 1
b. DS
0 1 0
0 0 2
0 0 0
c. detT 0. d. No.
8. a. DB
0 0 0 0
−3 0 0 0
3 −2 0 0
−4 9 2 − 1
2 0
b. DS
0 1 0 0
0 0 2 0
0 0 0 3
0 0 0 0
c. detT 0. d. No.
9. a. The members of B/ are non-zero, non-parallel linear combinations of sinx and cosx.
b. 3 −1
−1 3 c. TB
5 3 3 −4 3 − 6
4 3 6 −11 − 3 3 d. detT 2.
e. Yes; T−1 B
− 11 2 − 3
2 3 2 3 3
−2 3 − 3 5 2 3
2 3
f. 0 −1
1 0
Selected Answers to the Exercises 79
g. detD 1. h. Yes. D−1 B
0 1
−1 0 .
10. a. DB
−2 1 0
0 −2 2
0 0 −2
. b. detD −8 c. D−1 B
− 1 2 − 1
4 − 1
4
0 − 1 2 − 1
2
0 0 − 1 2
.
6.7 Exercises
1. a. TS
0 −5 −14
0 3 −10
0 0 14
b. detT 0; c. p − 3 − 14 d. 0, 3, 14
e. EigT, 0 Span1; EigT, 3 Span−5 3x; EigT, 14 Span52 70x − 77x2 f. TB Diag0, 3, 14, where B 1,−5 3x, 52 70x − 77x
2.
2. a. TS
4 5 −8 0
0 6 14 −24
0 0 14 27
0 0 0 28
b. detT 9408; c.
p − 4 − 6 − 14 − 28 d. 4, 6, 14, 28 e. EigT, 4 Span1; EigT, 6 Span5 2x; EigT, 14 Span3 70x 40x2; EigT, 28 Span−757 168x 2376x2 1232x3;
f. TB Diag4, 6, 14, 28, where B 1, 5 2x, 3 70x 40x2,−757 168x 2376x2 1232x3.
3. a. TS 0 −5
5 0 b. detT 25; c. p 2 25
d. The eigenvalues are imaginary, so . . . e. there are no eigenvectors for T, and consequently, . . . f. T is not diagonalizable.
4. a. DS
−1 0 0
0 2 0
0 0 5
b. detD −10; c. p 1 − 2 − 5
d. −1, 2, 5; f. EigD,−1 Spane−x; EigD, 2 Spane2x; EigD, 5 Spane5x; g. DS is already diagonal, so it is diagonalizable.
80 Selected Answers to the Exercises
5. a. DS
3 1 0
0 3 2
0 0 3
b. detD 27; c. p − 33 d. 3
e. EigD, 3 Spane3x f. D is not diagonalizable. 6. a. −1 for both sinx and cosx. b. The eigenvalue of ex is 2. c. e x d. −2
e. It has the same eigenvalue, −2. f. The common eigenvalue is 1. 7. TB Diag5,−1, 4, where B 1, 1 − 3x, 3 6x 5x
2
8. TS
3 2 1
−5 0 −1
0 1 −1
4 0 0
0 −7 0
0 0 3
− 1 12
− 1 4
1 6
5 12
1 4
1 6
5 12
1 4
− 5 6
− 67 12
− 23 4
− 17 6
5 12
17 4
− 5 6
− 25 6
− 5 2
4 3
9. There are 366 equivalence classes, including February 29.
Selected Answers to the Exercises 81
Chapter Seven Exercises
7.1 Exercises 9. −46 10. − 22/5 11. 16 12. − 276 13. 22 14. −72 15. 38 16. − 10, 892 17.
1/2 18. 0 19. rx x 1x − 1x − 2x − 4 or any scalar multiple thereof. 20. No. 21. No. 22. Yes. 23. −/4 30. b. removable discontinuity 31. a. Further hint: since the series ∑an converges, the terms an must converge to 0, so
therefore if n is large enough, |an | 1. c. (use geometric series formula) 1/5; d. −1/16.
7.2 Exercises 1. 279 ; u/ 279 2. 341 ; u/ 341 3. 131 ; u/ 131 4. 354 ; px/ 354 5.
1802 ; px/ 1802 6. cos−1 312/ 8051 and du, v 8 7. cos−1 16/ 17510 and
du, v 241 8. cos−111/15 and du, v 65 9. cos−1 −298/ 131, 334 and
du, v 1321 10. cos−1 −10892/ 120816920 and du, v 47290 11. 8/ 15 ; 12. cos 2/ 2 − 4 , so ≈ 0. 6 radians; 13. 49x2 25y2 1 is an ellipse
(left, below):
0.1250.10.0750.050.0250-0.025-0.05-0.075-0.1-0.125
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025 0
-0.025
-0.05
-0.075
-0.1
-0.125
-0.15
-0.175
-0.2
x
y
x
y
6.2553.752.51.250-1.25-2.5-3.75-5-6.25
5
4
3
2
1
0
-1
-2
-3
-4
-5
x
y
x
y
14. x 2
49
y2
25 1 is an ellipse (above, right).
15. 4x2 y2 25z2 1 is an ellipsoid with vertices 1/2, 0, 0, 0,1, 0, 0, 0,1/5 16. 7342 17. No. 19 15, so the conditions violate the Cauchy-Schwarz Inequality. 18. ‖u‖ 13 and
‖v‖ 5. 37. You get an isosceles triangle.
7.3 Exercises
1. 1 3 ⟨1, 1,−1, 1
6 ⟨2,−1, 1, 1
2 ⟨0, 1, 1
82 Selected Answers to the Exercises
2. 1 2 ⟨1, 0, 1, 1
3 ⟨−1, 1, 1, 1
6 ⟨1, 2,−1
3. 1 12
⟨1, 1,−1, 1 24
⟨2,−1, 1, 1 120
⟨0, 3, 5 ; different answer.
4. 1 7 ⟨1, 1,−1, 1
581 ⟨6,−7,−8, 1
4980 ⟨15, 24,−20 ; different answer.
5. 1 5 ⟨1, 1,−1, 1
5 ⟨2,−3, 3,⟨1,−1, 2 ; different answer.
6. 1 2 ⟨1, 0, 1, 1
11 ⟨2,−1, 0, 1
22 ⟨−5, 8,−11
7. 1 2 ⟨1,−1, 1,−1, 1
2 11 ⟨5,−1,−3, 3, 1
330 ⟨7, 14,−2,−9, 1
30 ⟨1, 2, 4, 3
8. 1 3 ⟨1,−1, 0, 1, 1
15 ⟨1, 2,−3, 1, 1
3 10 ⟨7, 4, 4,−3, 1
3 2 ⟨−1, 2, 2, 3
9. 1 14
⟨1,−1, 1,−1, 1 2198
⟨19,−5,−9, 9, 1 125286
⟨33, 264,−90,−67, 1 2 399
⟨3, 24, 16
10. 1 11
⟨1,−1, 0, 1, 1 473
⟨1, 10,−11, 1, 1 21930
⟨57, 54, 18,−29, 1 1020
⟨−3, 24, 8, 6
different answer.
11. 1 17
x2, 1 6 17
7x2 17x, 1 2 x2 x − 2
12. 1 30
x2 1, 1 330
8x2 15x − 7, 1 99
4x2 2x − 9
13. 5 x2, 3 5x2 − 4x, 10x2 − 12x 3
14. 1 3
, 1 42
3x 1, 1 126
7x2 8x − 9 ; different answer.
15. 7 x3, 5 6x2 − 7x3, 3 21x3 − 30x2 10x,−35x3 60x2 − 30x 4 16. ⟨uS − 3 , 3 6 /2,−3 2 /2 , and ⟨vS −2 3 ,− 6 /2, 13 2 /2
17. ⟨uS 3 2
,− 5 3
,− 7 6
, and ⟨vS 5 2
, 16 3
,− 1 6
18. ⟨uS − 15 12
, 39 24
,− 45 120
, and ⟨vS − 11 12
,− 25 24
, 195 120
19. ⟨uS 11 7 , 3 83
581 ,− 102 83
1245 , and ⟨vS 12 7 ,− 85 83
581 ,− 98 83
1245
20. ⟨uS 1 5
, 12 5
,−3 , and ⟨vS − 12 5
,− 34 5
, 13
21. ⟨uS − 5 2
, 16 11
,− 7 22
, and ⟨vS 15 2
,− 59 11
,− 1 22
22. ⟨uS − 1 2
, 3 2 11
, 145 330
, −5 30
, and ⟨vS 17 2
, −3 2 11
, 20 330
, 20 30
23. ⟨uS − 7 3
3 , 17 15
15 , 49 30
10 ,− 7 6
2 , and
Selected Answers to the Exercises 83
⟨vS 4 3
3 ,− 23 15
15 , 32 15
10 ,− 2 3
2
24. ⟨uS 12 7
14 , 18 1099
2198 , 688 20881
125286 ,− 5 133
399 , and
⟨vS 61 14
14 , 39 2198
2198 ,− 92 20881
125286 , 40 133
399
25. ⟨uS −18
11 , 114
473 , 1596
21930 ,− 84
1020 , and
⟨vS 4 11
,− 249 473
, 1932 21930
,− 48 1020
26. ⟨uS −56 17 /17,−3 17 /17, 5 , and ⟨vS 73 17 /17,−48 17 /17, 4
27. ⟨uS −78
30 , 36
330 , 6
99 , and ⟨vS
82 30
, −184 330
, 117 99
28. ⟨uS − 41 30
5 , 3 2
3 ,− 5 3
, and ⟨vS − 67 30
5 , 11 6
3 ,− 4 3
29. ⟨uS −22
3 , 44
42 , −18
126 , and ⟨vS
9 3
, −132 42
, 54 126
30. ⟨uS 461 294
7 , 52 105
5 , 29 210
3 ,− 3 35
, and
⟨vS 433 1470
7 ,− 1 21
5 ,− 11 210
3 , 1 7
31. a. 0
2 sinxcosxdx
0
2 sin2xcosxdx
0
2 sin2xsinxdx 0,
0
2 sin22xdx
0
2 sin2xdx
0
2 cos2xdx
b. 1
sinx, 1
cosx, 1
sin2x c. ⟨uS ⟨2, 7,−3, and
⟨vS ⟨5,−2, 1 32. B is linearly dependent, because w 3 is in the Span of w 1, w 2. 34. px x 3x − 1, qx x 3x − 4, rx x − 1x − 4 is a possible answer.
7.4 Exercises 1. ⟨1,−1, 1 2. ⟨1, 1, 0,⟨−3, 0, 1 3. ⟨15,−12, 20 4. ⟨5, 4, 0,⟨−3, 0, 2 5. ⟨1, 0, 1, 0,⟨−1,−2, 0, 1 6. ⟨3, 0, 4, 0,⟨−3,−24, 0, 2 7. ⟨−9,−24,−8, 2 8.
⟨2, 3,−3 9. ⟨3, 5, 0,⟨1, 0, 1 10. 7x2 17x,−5x2 17 11. 5x2 7x − 9 12. 224 888x − 251x2, 414 − 3827x 251x3 13. 1 − 5x, 8 − 5x2, 7 − 5x3 14. 25x2 − 17x, 50x2 − 17
15. for W : 1 3 ⟨1, 1,−1, 1
6 ⟨2,−1, 1 ; for W : 1
2 ⟨0, 1, 1
16. for W : 1 2 ⟨1, 0, 1 ; for W : , 1
3 ⟨−1, 1, 1, 1
6 ⟨1, 2,−1
17. for W : 1 2 3
⟨1, 1,−1 ; for W : 1 24
⟨2,−1, 1, 1 120
⟨0, 3, 5
18. for W : 1 7 ⟨1, 1,−1, 1
581 ⟨6,−7,−8 ; for W : 1
4980 ⟨15, 24,−20
84 Selected Answers to the Exercises
19. for W : 1 2 ⟨1, 0, 1, 1
11 ⟨2,−1, 0 ; for W : 1
22 ⟨−5, 8,−11
20. for W : 1 2 ⟨1,−1, 1,−1, 1
2 11 ⟨5,−1,−3, 3 ; for
W : 1 330
⟨7, 14,−2,−9, 1 30
⟨1, 2, 4, 3
21. for W : 1 3 ⟨1,−1, 0, 1 ; for
W : 1 15
⟨1, 2,−3, 1, 1 3 10
⟨7, 4, 4,−3, 1 3 2
⟨−1, 2, 2, 3
22. for W : 1 14
⟨1,−1, 1,−1, 1 2198
⟨19,−5,−9, 9, 1 125286
⟨33, 264,−90,−67 ;
for W : 1 2 399
⟨3, 24, 16, 6
23. for W : 1 17
x2 ; for W : 1 6 17
7x2 17x, 1 2 x2 x − 2
24. for W : 1 30
x2 1, 1 330
8x2 15x − 7 ; for W : 1 99
4x2 2x − 9
25. for W : 5 x2, 3 5x2 − 4x ; for W : 10x2 − 12x 3
26. Start with ⟨5,−2, 0, e1, e3; for V : 1 29
⟨5,−2, 0, 1 29
⟨2, 5, 0, e3 ;
for W : 1 29
⟨5,−2, 0 ; for W : 1 29
⟨2, 5, 0, e3
27. Start with ⟨5,−2, 0, e1, e3; for V : ⟨5,−2, 0/ 120 ,⟨1, 2, 0/ 24 ,⟨0, 0, 1/ 2 ; for W : ⟨5,−2, 0/ 120 ; for W : ⟨1, 2, 0/ 24 ,⟨0, 0, 1/ 2
28. Start with ⟨1,−1, 0, 1, ⟨1, 0,−3, 1, e1, e2; For V : ⟨1,−1, 0, 1/ 3 ,⟨1, 2,−9, 1/ 87 ,⟨19, 9, 3,−10/ 551 ,⟨0, 3, 1, 3/ 19 for W : ⟨1,−1, 0, 1/ 3 ,⟨1, 2,−9, 1/ 87 ; for W : ⟨19, 9, 3,−10/ 551 ,⟨0, 3, 1, 3/ 19
29. Start with ⟨1,−1, 0, 1, ⟨1, 0,−3, 1, e1, e2;
For V : 1 11
⟨1,−1, 0, 1, 1 3377
⟨1, 10,−33, 1, 1 2 59865
⟨195, 108, 12,−112,
1 104 409 309 290
⟨−97 825, 115 898, 57 222, 84 533
for W : 1 11
⟨1,−1, 0, 1, 1 3377
⟨1, 10,−33, 1 ; for W :
1 2 59865
⟨195, 108, 12,−112, 1 104 409 309 290
⟨−97 825, 115 898, 57 222, 84 533
30. Start with x2 5x, x2, 1; for V : 1 72
x2 5x, 1 8 x2 x, 1
2 x2 x − 2 ;
Selected Answers to the Exercises 85
for W : 1 72
x2 5x ; for W : 1 8 x2 x, 1
2 x2 x − 2
31. Start with x2 − 3x, x, 1; for V : 10 17 x2 − 3x, 6
17 15x2 − 11x, 10x2 − 12x 3 ;
for W : 10 17 x2 − 3x, 6
17 15x2 − 11x ; for W : 10x2 − 12x 3
33. w 1 ⟨2,−5/2, 5/2, and w 2 ⟨0,−3/2,−3/2 34. w 1 ⟨5/2, 0, 5/2, and w 2 ⟨−11/2, 5, 11/2
35. w 1 − 54 ⟨1, 1,−1, and w 2 14
⟨13,−11,−1 36. w 1 122 ⟨−71, 118, 165, and
w 2 − 122 ⟨−5, 8,−11
37. w 1 111 , 2
11 ,− 5
11 , 5
11 , and w 2 3211
, 64 11
,− 17 11
,− 49 11
38. w 1 43 ,− 4
3 , 0, 4
3 , and w 2 113
,− 2 3
, 7,− 13 3
39. w 1 − 15133 ,− 120 133
,− 80 133
,− 30 133
, and w 2 414133 , 918 133
,− 186 133
,− 502 133
40. w 1 −56x2/17, and w 2 39x2/17 2x − 5 41. w 1 493 x2 − 22x, and
w 2 − 403 x2 16x − 4
7.5 Exercises 1. a. u | v −100; b. ‖u‖ 3 11 ; c. ‖v‖ 353 ; d. du, v 2 163
e. cos − 100 3 11 353
2. a. u | v −123; b. ‖u‖ 38 ; c. ‖v‖ 429 ; d. du, v 713 e. cos − 123
38 429 3. a. u | v 78; b. ‖u‖ 6 5 ; c. ‖v‖ 305 ; d. du, v 329
e. cos 13 5 61
4. a. u | v −212; b. ‖u‖ 210 ; c. ‖v‖ 465 ; d. du, v 1099 ; e. cos − 212
97650 5. a. u | v 386
15 ; b. ‖u‖ 1
15 4245 ; c. ‖v‖ 2
5 230 ; d. du, v 1
5 105 ;
e. cos 193 39054
6.
1 0 0
0 1 2
− 1 2
0 − 1 2
1 2
7.
1 2
0 1 2
0 0 0 1 2
0 1 2
8.
9 11
− 4 11
− 1 11
1 11
− 4 11
3 11
− 2 11
2 11
− 1 11
− 2 11
5 11
− 5 11
1 11
2 11
− 5 11
5 11
86 Selected Answers to the Exercises
9.
1 3
− 1 3
0 1 3
− 1 3
1 3
0 − 1 3
0 0 0 0 1 3
− 1 3
0 1 3
10.
2 3
− 1 3
0 1 3
− 1 3
2 3
0 1 3
0 0 1 0 1 3
1 3
0 2 3
11. One way is to apply Gram-Schmidt to ⟨3, 5, 0,⟨7, 0, 5; we get 58 83
15 83
35 83
15 83
74 83
− 21 83
35 83
− 21 83
34 83
12.
1 0 0
0 16 65
28 65
0 28 65
49 65
21. c. fx −7x4 5x2 − 1, and gx 8x5 − 2x3 6x
7.6 Exercises
1. −8/17 15/17
15/17 8/17 is improper, while
−8/17 −15/17
15/17 −8/17 is proper.
2. 20/29 −21/29
−21/29 −20/29 is improper, while
20/29 −21/29
21/29 20/29 is proper.
3.
1 3
2 6
0
1 3 − 1
6 1 2
− 1 3
1 6
1 2
(improper). 4.
1 2
5 2 11
7 330
1 30
− 1 2 − 1
2 11 14 330
2 30
1 2 − 3
2 11 −2 330
4 30
− 1 2
3 2 11
−9 330
3 30
,
(proper)
5. b. Q −20/29 21/29
21/29 20/29 and Q/
15/17 −8/17
8/17 15/17 c. Q is improper and Q/ is
proper.
d. QQ/ − 132
493 475 493
475 493
132 493
. e. QQ/ is improper. f. CB,B/ − 132
493 475 493
475 493
132 493
g. CB,B/ is improper.
Selected Answers to the Exercises 87
6. There are 2n possible combinations. 7. There are n! such rearrangements.
15. g. 1 a2b2
⟨−b, a, 0, 1 a2b2
⟨−ac,−bc, a2 b2
i) 1 13
⟨2, 3, 0, 1 7 13
⟨−18, 12, 13, 1 7 ⟨3,−2, 6
7.7 Exercises 1. Q is proper.
Note: other answers are possible in the following if the eigenspace has dimension 2 or bigger.
2.
0 0 0
0 1 0
0 0 3
;
1/ 3 −1/ 2 1/ 6
1/ 3 0 −2/ 6
1/ 3 1/ 2 1/ 6
3.
−1 0 0
0 0 0
0 0 3
;
−1/ 2 1/ 3 1/ 6
0 −1/ 3 2/ 6
1/ 2 1/ 3 1/ 6
4.
−1 0 0
0 2 0
0 0 2
;
1/ 3 −1/ 2 −1/ 6
1/ 3 0 2/ 6
1/ 3 1/ 2 −1/ 6
5.
−2 0 0
0 2 0
0 0 3
;
−1/ 2 1/ 2 0
0 0 1
1/ 2 1/ 2 0
6.
−5 0 0
0 −5 0
0 0 10
;
−1/ 2 −1/ 6 1/ 3
0 2/ 6 1/ 3
1/ 2 −1/ 6 1/ 3
7.
2 − 2 0 0
0 2 0
0 0 2 2
;
1/2 1/ 2 1/2
− 2 /2 0 2 /2
1/2 −1/ 2 1/2
8.
1 0 0
0 1 0
0 0 4
;
−1/ 2 −1/ 6 1/ 3
0 2/ 6 1/ 3
1/ 2 −1/ 6 1/ 3
9.
4 0 0
0 4 0
0 0 7
;
−1/ 2 −1/ 6 1/ 3
0 2/ 6 1/ 3
1/ 2 −1/ 6 1/ 3
88 Selected Answers to the Exercises
10.
−1 0 0
0 8 0
0 0 8
;
1/ 3 −1/ 2 −1/ 6
1/ 3 0 2/ 6
1/ 3 1/ 2 −1/ 6
11.
4 0 0
0 5 0
0 0 6
;
−1/ 2 0 1/ 2
0 1 0
1/ 2 0 1/ 2
12.
−7 0 0
0 −7 0
0 0 −4
;
1/ 2 −1/ 6 1/ 3
0 2/ 6 1/ 3
−1/ 2 −1/ 6 1/ 3
13.
−1 0 0
0 1 0
0 0 5
;
1/ 2 1/ 2 0
0 0 1
−1/ 2 1/ 2 0
14.
−5 0 0 0
0 −5 0 0
0 0 −5 0
0 0 0 −1
;
−1/ 2 −1/ 6 −1/ 12 1/2
0 0 3/ 12 1/2
1/ 2 −1/ 6 −1/ 12 1/2
0 2/ 6 −1/ 12 1/2
15.
−7 0 0 0
0 −7 0 0
0 0 −7 0
0 0 0 5
;
−1/ 2 −1/ 6 −1/ 12 1/2
0 0 3/ 12 1/2
1/ 2 −1/ 6 −1/ 12 1/2
0 2/ 6 −1/ 12 1/2
16.
−4 0 0 0
0 −2 0 0
0 0 0 0
0 0 0 2
;
0 −1/2 −1/ 2 1/2
−1/ 2 1/2 0 1/2
0 −1/2 1/ 2 1/2
1/ 2 1/2 0 1/2
17.
4 0 0 0
0 4 0 0
0 0 6 0
0 0 0 6
;
−1/ 2 0 1/ 2 0
0 −1/ 2 0 1/ 2
0 1/ 2 0 1/ 2
1/ 2 0 1/ 2 0
Selected Answers to the Exercises 89
18.
−7 0 0 0
0 −7 0 0
0 0 −3 0
0 0 0 −3
;
−1/ 2 0 1/ 2 0
0 −1/ 2 0 1/ 2
0 1/ 2 0 1/ 2
1/ 2 0 1/ 2 0
19.
−5 0 0 0
0 −2 0 0
0 0 2 0
0 0 0 5
;
−1/ 2 0 0 1/ 2
0 1/ 2 −1/ 2 0
0 1/ 2 1/ 2 0
1/ 2 0 0 1/ 2
20.
−5 0 0 0 0
0 −2 0 0 0
0 0 2 0 0
0 0 0 5 0
0 0 0 0 7
;
−1/ 2 0 0 1/ 2 0
0 1/ 2 −1/ 2 0 0
0 0 0 0 1
0 1/ 2 1/ 2 0 0
1/ 2 0 0 1/ 2 0
21.
6 0 0 0 0
0 6 0 0 0
0 0 7 0 0
0 0 0 8 0
0 0 0 0 8
;
0 1/ 2 0 0 1/ 2
1/ 2 0 0 1/ 2 0
0 0 1 0 0
−1/ 2 0 0 1/ 2 0
0 −1/ 2 0 0 1/ 2
22.
6 0 0 0 0
0 6 0 0 0
0 0 1 0 0
0 0 0 8 0
0 0 0 0 8
;
0 1/ 2 0 0 1/ 2
1/ 2 0 0 1/ 2 0
0 0 1 0 0
−1/ 2 0 0 1/ 2 0
0 −1/ 2 0 0 1/ 2 There is exactly one different eigenvalue between the matrices in Exercise 21 and 22, but the diagonalizing orthogonal matrix is the same for both. This is explained further in Exercises 26 and 27.
23. a) Eigenvalues a 2b, with multiplicity 1, and a − b, with multiplicity 2. b) Eigenvalues a 3b, with multiplicity 1, and a − b, with multiplicity 3.
24. a) Eigenvalues c1 and c2. b) Eigenvalues ci, all with multiplicity 1. 25. b) Eigenvalues c1,c2,…ck−1, ck, all with multiplicity 1, where k n 1/2. 26. b) Eigenvalues a b, each with multiplicity n/2. 27. c) Eigenvalues a, with multiplicity 1, and a b, each with multiplicity n − 1/2.
d) Eigenvalues b, with multiplicity 1, and a b, each with multiplicity n − 1/2.
90 Selected Answers to the Exercises
30. Q
− 1 5
5 1 15
30 − 1 3
6 2 5
5 1 30
30 − 1 6
6
0 1 6
30 1 6
6
; D
−12 0 0
0 −12 0
0 0 18
31. Q
− 1 5
5 1 15
30 − 1 3
6 2 5
5 1 30
30 − 1 6
6
0 1 6
30 1 6
6
; D
−15 0 0
0 −15 0
0 0 9
32. Q
1 2
2 − 1 3
3 1 6
6 1 2
2 1 3
3 − 1 6
6
0 1 3
3 1 3
6
; D
−2 0 0
0 −2 0
0 0 4
33. Q
− 1 14
14 3 10
10 1 35
35 3 14
14 1 10
10 − 3 35
35 1 7
14 0 1 7
35
; D
−35 0 0
0 −21 0
0 0 −21
34. Q
1 2
2 − 1 3
3 1 6
6 1 2
2 1 3
3 − 1 6
6
0 1 3
3 1 3
6
; D
−15 0 0
0 −15 0
0 0 9
35. Q
− 1 14
14 3 10
10 1 35
35 3 14
14 1 10
10 − 3 35
35 1 7
14 0 1 7
35
; D
−98 0 0
0 56 0
0 0 56
36. Q
− 1 14
14 3 10
10 1 35
35 3 14
14 1 10
10 − 3 35
35 1 7
14 0 1 7
35
; D
−6 0 0
0 8 0
0 0 8
37. Q
− 1 14
14 3 10
10 1 35
35 3 14
14 1 10
10 − 3 35
35 1 7
14 0 1 7
35
; D
−4 0 0
0 10 0
0 0 10
38. Q
1 2
2 − 1 3
3 1 6
6 1 2
2 1 3
3 − 1 6
6
0 1 3
3 1 3
6
; D
−12 0 0
0 −12 0
0 0 30
Selected Answers to the Exercises 91
39. Q
− 3 83
83 − 12 2905
8715 1 11
22 6 385
2310 5 83
83 − 23 8715
8715 − 3 22
22 23 2310
2310 7 83
83 1 8715
8715 3 22
22 − 1 2310
2310
0 1 105
8715 0 1 105
2310
;
D
−42 0 0 0
0 −42 0 0
0 0 63 0
0 0 0 63
40. Q
3 11
11 − 1 77
231 1 10
10 1 70
210
− 1 11
11 1 231
231 3 10
10 − 1 210
210 1 11
11 10 231
231 0 − 1 21
210
0 1 21
231 0 1 21
210
; D
−63 0 0 0
0 −63 0 0
0 0 21 0
0 0 0 21
41. Q
3 11
11 − 1 77
231 1 10
10 1 70
210
− 1 11
11 1 231
231 3 10
10 − 1 210
210 1 11
11 10 231
231 0 − 1 21
210
0 1 21
231 0 1 21
210
; D
0 0 0 0
0 0 0 0
0 0 21 0
0 0 0 21
42. Q
13 203
203 − 1 203
1015 1 42
42 1 42
210
− 5 203
203 − 1 145
1015 5 42
42 1 30
210 3
203 203 2
203 1015 2
21 42 − 1
21 210
0 1 35
1015 0 1 35
210
; D
0 0 0 0
0 0 0 0
0 0 35 0
0 0 0 35
43. Q
3 58
58 − 7 899
899 − 1 186
434 1 3
7 7 58
58 3 899
899 1 434
434 − 1 7
7
0 1 31
899 − 1 651
434 2 21
7
0 0 1 21
434 1 21
7
; D
−63 0 0 0
0 −63 0 0
0 0 −63 0
0 0 0 0
44. Q
− 1 7
14 8 287
574 − 1 1148
574 − 2 15
15 1 420
210
− 3 14
14 − 11 574
574 − 1 164
574 1 15
15 1 60
210 1 14
14 − 1 574
574 − 23 1148
574 − 1 15
15 23 420
210
0 1 41
574 − 3 574
574 1 5
15 1 70
210
0 0 1 28
574 0 1 28
210
;
92 Selected Answers to the Exercises
D
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 56 0
0 0 0 0 56
7.8 Exercises
1. The rref of
3 −1 −6 2
−2 1 3 −2
5 −2 −9 1
is
1 0 −3 0
0 1 −3 0
0 0 0 1
, so the system is inconsistent.
C
3 −1
−2 1
5 −2
, CC 38 −15
−15 6 , x
0
−1 , x1 ⟨0,−1, 0,
the nullspace has basis ⟨3, 3, 1, 0,
Ax1
1
−1
2
b1, projW
2 3
1 3
1 3
1 3
2 3 − 1
3
1 3 − 1
3 2 3
, b − b1 1
−1
−1
,
the common error is 3 .
2. The rref of
1 1 −1 −2
1 −2 5 9
2 −1 4 5
2 1 0 4
is
1 0 1 0
0 1 −2 0
0 0 0 1
0 0 0 0
, so the system is inconsistent.
C
1 1
1 −2
2 −1
2 1
, CC 10 −1
−1 7 , x
154 69
− 185 69
, x1 15469 ,− 185 69
, 0 ,
the nullspace has basis ⟨−1, 2, 1,
Ax1
− 31 69
524 69
493 69
41 23
b1, projW 169
19 −14 5 27
−14 43 29 −9
5 29 34 18
27 −9 18 42
, b − b1
− 107 69
97 69
− 148 69
51 23
,
the common error is 1 69
66171 ≈ 3. 7281
Selected Answers to the Exercises 93
3. The rref of
3 −15 −6 2 28
−2 10 4 −4 −26
5 −25 −10 −1 13
is
1 −5 −2 0 0
0 0 0 1 0
0 0 0 0 1
,
C
3 2
−2 −4
5 −1
, CC 38 9
9 21 , x
966 239
1259 239
, x1 966239 , 0, 0, 1259 239
,
the nullspace has basis ⟨5, 1, 0, 0,⟨2, 0, 1, 0,
Ax1
5416 239
− 6968 239
3571 239
b1, projW 1717
233 −286 176
−286 548 104
176 104 653
,
b − b1
1276 239
754 239
− 464 239
,
the common error is 58 239
717 ≈ 6. 4981.
4. The rref of
3 −2 19 4 38
4 −1 22 −3 5
−1 5 −15 2 −28
1 2 1 4 2
is
1 0 5 0 0
0 1 −2 0 0
0 0 0 1 0
0 0 0 0 1
, so the system is
inconsistent.
C
3 −2 4
4 −1 −3
−1 5 2
1 2 4
, CC
27 −13 2
−13 34 13
2 13 45
, x
34762 14165
− 98068 14165
54801 14165
,
x1 3476214165 ,− 98068 14165
, 0, 54801 14165
, the nullspace has basis ⟨−5, 2, 1, 0,
Ax1
519626 14165
72713 14165
− 83100 2833
11566 2833
b1, projW 128330
22089 632 −6320 9875
632 28266 640 −1000
−6320 640 21930 10000
9875 −1000 10000 12705
,
b − b1
18644 14165
− 1888 14165
3776 2833
− 5900 2833
, the common error is 236 14165
28330 ≈ 2. 8043.
94 Selected Answers to the Exercises
5. The rref of
4 −3 1 11
−2 0 −5 −9
3 1 2 5
−1 5 6 −7
0 3 2 −4
is
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
, so the system is inconsistent.
C
4 −3 1
−2 0 −5
3 1 2
−1 5 6
0 3 2
, CC
30 −14 14
−14 44 35
14 35 70
, x
1991 1399
− 2804 1399
9265 9793
x1,
the nullspace is 04 , so x1 is a unique solution,
Ax1
123897 9793
− 74199 9793
40713 9793
− 56487 9793
− 40354 9793
b1,
projW 19793
11288 −5998 5510 −4666 −2358
−5998 13400 1934 −5818 3054
5510 1934 13659 391 6834
−4666 −5818 391 13731 4962
−2358 3054 6834 4962 6680
,
b − b1
− 16 174 9793
− 13 938 9793
8252 9793
− 12 064 9793
1182 9793
, the common error is 6 9793
18636079 ≈ 2. 6449.
9. C
7 −4
3 0
0 3
(other choices are possible), so CC 58 −28
−28 25 , and
Selected Answers to the Exercises 95
proj
7 −4
3 0
0 3
58 −28
−28 25
−1 7 3 0
−4 0 3
65 74
21 74 − 6
37
21 74
25 74
14 37
− 6 37
14 37
29 37
7.9 Exercises
1. 1 5
5 2 5
5 2 5
5 − 1 5
5
5 13 5
5
0 1 5
5 2.
1 5
5 − 2 5
5 2 5
5 1 5
5
5 0
0 5
3.
1 2
2 − 1 6
2
0 2 3
2 1 2
2 1 6
2
2 − 1 2
2
0 3 2
2 4.
1 11
11 − 9 682
682
0 1 31
682
− 1 11
11 9 682
682 3
11 11 3
341 682
11 − 2 11
11
0 1 11
682
5.
1 2
2 − 1 6
2 − 2 3
0 2 3
2 − 1 3
1 2
2 1 6
2 2 3
2 − 1 2
2 3 2
2
0 3 2
2 7 6
2
0 0 5 3
6.
11 11
− 9 682 682
43 2418 2418
0 682 31
2 2418 403
− 11 11
9 682 682
19 2418 2418
3 11 11
3 682 341
− 4 2418 1209
11 − 2 11 11
− 4 11 11
0 682 11
25 682 682
0 0 2418 62
7.
11 11
− 9 682 682
43 2418 2418
39 39
0 682 31
2 2418 403
39 13
− 11 11
9 682 682
19 2418 2418
− 5 39 39
3 11 11
3 682 341
− 4 2418 1209
− 2 39 39
11 − 2 11 11
− 4 11 11
8 11 11
0 682 11
25 682 682
8 682 341
0 0 2418 62
10 2418 1209
0 0 0 5 39
39
96 Selected Answers to the Exercises
8.
7 7
− 231 77
4 33 33
0 2 231
33 33 22
− 7 7
231 77
33 22
7 7
4 231 231
− 7 33 66
2 7 7
231 231
33 66
7 − 4 7 7
− 4 7 7
0 231 7
4 231 77
0 0 2 33
11
9.
7 7
− 231 77
4 33 33
30 10
0 2 231
33 33 22
30 60
− 7 7
231 77
33 22
− 30 20
7 7
4 231 231
− 7 33 66
30 12
2 7 7
231 231
33 66
− 7 30 60
7 − 4 7 7
− 4 7 7
4 7 7
0 231 7
4 231 77
16 231 231
0 0 2 33
11 − 33
11
0 0 0 30 3
We will show in the answers below only the QR-decomposition of C, the matrix consisting of the linearly independent columns of A, and how to obtain the unique solution x R−1Qb to the normal system. The rest of the solution is shown in the answers to 7.8:
10. C
3 38 38
7 114 114
− 38 19
4 114 57
5 38 38
− 114 114
38 − 15 38 38
0 114 38
;
38 38
5 114 38
0 114 3
3 38 38
− 38 19
5 38 38
7 114 114
4 114 57
− 114 114
2
−2
1
0
−1 x1
11. C
10 10
11 690 690
10 10
− 19 690 690
10 5
− 4 690 345
10 5
2 690 115
10 − 10 10
0 690 10
;
10 10
690 690
0 690 69
10 10
10 10
10 5
10 5
11 690 690
− 19 690 690
− 4 690 345
2 690 115
−2
9
5
4
154 69
− 185 69
x
Selected Answers to the Exercises 97
12. C
3 −15 −6 2
−2 10 4 −4
5 −25 −10 −1
3 38 38
49 27246 27246
− 38 19
− 67 27246 13623
5 38 38
− 83 27246 27246
38 9 38
38
0 27246
38
;
38 38
− 3 27246 9082
0 27246 717
3 38 38
− 38 19
5 38 38
49 27246 27246
− 67 27246 13623
− 83 27246 27246
28
−26
13
966 239
1259 239
13. C
3 3
− 5 2247 2247
3039 21219170 21219170
4 3 9
25 2247 6741
− 1409 21219170 10609585
− 3 9
122 2247 6741
− 15 21219170 2121917
3 9
67 2247 6741
401 21219170 4243834
3 3 − 13 3 9
2 3 9
0 2247
9 377 2247
6741
0 0 21219170
749
;
3 9
13 2247 6741
− 237 21219170 21219170
0 3 2247
749 − 377 21219170
21219170
0 0 21219170
28330
3 3
4 3 9
− 3 9
3 9
− 5 2247 2247
25 2247 6741
122 2247 6741
67 2247 6741
3039 21219170 21219170
− 1409 21219170 10609585
− 15 21219170 2121917
401 21219170 4243834
38
5
−28
2
34762 14165
− 98068 14165
54801 14165
x1
14. A C
2 30 15
− 17 8430 8430
219 5503666 5503666
− 30 15
− 7 8430 4215
− 852 5503666 2751833
30 10
6 8430 1405
− 579 5503666 2751833
− 30 30
34 8430 4215
405 5503666 2751833
0 3 8430
562 − 745 5503666
5503666
30 − 7 30 15
7 30 15
0 8430 15
623 8430 8430
0 0 5503666
562
;
98 Selected Answers to the Exercises
30 30
7 8430 8430
− 79 5503666 786238
0 8430 562
− 89 5503666 786238
0 0 5503666
9793
2 30 15
− 30 15
30 10
− 30 30
0
− 17 8430 8430
− 7 8430 4215
6 8430 1405
34 8430 4215
3 8430 562
219 5503666 5503666
− 852 5503666 2751833
− 579 5503666 2751833
405 5503666 2751833
− 745 5503666 5503666
11
−9
5
−7
−4
1991 1399
− 2804 1399
9265 9793
x1
Selected Answers to the Exercises 99
