MATH1251 Mathematics for Actuarial Studies and Finance …web.maths.unsw.edu.au/~potapov/1251_2014/math1251AlgProblems.pdf · Chapter 6 COMPLEX NUMBERS 6.1 A review of number systems

MATH1251Mathematics for Actuarial Studies

and Finance 1B

ALGEBRA PROBLEMS

Semester 2 2014

Copyright 2014 School of Mathematics and Statistics, UNSW

Contents

6 COMPLEX NUMBERS 1

6.1 A review of number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Introduction to complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The rules of arithmetic for complex numbers . . . . . . . . . . . . . . . . . . . . . . 16.4 Real parts, imaginary parts and complex conjugates . . . . . . . . . . . . . . . . . . 16.5 The Argand diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Polar form, modulus and argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Properties and applications of the polar form . . . . . . . . . . . . . . . . . . . . . . 16.8 Trigonometric applications of complex numbers . . . . . . . . . . . . . . . . . . . . . 16.9 Geometric applications of complex numbers . . . . . . . . . . . . . . . . . . . . . . . 16.10 Complex polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.11 Stability of discrete and continuous time systems . . . . . . . . . . . . . . . . . . . . 16.12 Appendix: A note on proof by induction . . . . . . . . . . . . . . . . . . . . . . . . . 16.13 Appendix: The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.14 Complex numbers and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

7 VECTOR SPACES 15

7.1 Definitions and examples of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Vector arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Subspaces of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 Linear combinations and spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.6 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.7 Coordinate vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.8 Further important examples of vector spaces . . . . . . . . . . . . . . . . . . . . . . 157.9 Data fitting and polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . 157.10 Appendix: A brief review of set and function notation . . . . . . . . . . . . . . . . . 157.11 Vector spaces and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15


8 LINEAR TRANSFORMATIONS 33

8.1 Introduction to linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Linear maps from Rn to Rm and m! n matrices . . . . . . . . . . . . . . . . . . . . 33

iii

8.3 Geometric examples of linear transformations . . . . . . . . . . . . . . . . . . . . . . 338.4 Subspaces associated with linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . 338.5 Further applications and examples of linear maps . . . . . . . . . . . . . . . . . . . . 338.6 Representation of linear maps by matrices . . . . . . . . . . . . . . . . . . . . . . . . 338.7 Matrix arithmetic and linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.8 One-to-one, onto and invertible linear maps and matrices . . . . . . . . . . . . . . . 338.9 Proof of the Rank-Nullity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.10 One-to-one, onto and inverses for functions . . . . . . . . . . . . . . . . . . . . . . . 338.11 Linear transformations and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


9 EIGENVALUES AND EIGENVECTORS 47

9.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.2 Eigenvectors, bases, and diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . 479.3 Eigenvalues of matrices with special structure . . . . . . . . . . . . . . . . . . . . . . 479.4 Applications of eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 479.5 Markov systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.6 Eigenvalues and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


ANSWERS TO SELECTED PROBLEMS 55

Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

PAST CLASS TESTS 79

iv

Chapter 6

COMPLEX NUMBERS

6.1 A review of number systems

6.2 Introduction to complex numbers

6.3 The rules of arithmetic for complex numbers

6.4 Real parts, imaginary parts and complex conjugates

6.5 The Argand diagram

6.6 Polar form, modulus and argument

6.7 Properties and applications of the polar form

6.8 Trigonometric applications of complex numbers

6.9 Geometric applications of complex numbers

6.10 Complex polynomials

6.11 Stability of discrete and continuous time systems

6.12 Appendix: A note on proof by induction

6.13 Appendix: The Binomial Theorem

6.14 Complex numbers and Matlab

1

2 CHAPTER 6. COMPLEX NUMBERS

Problems for Chapter 6

“Why,” said the Dodo, “the best way to explain it is to do it.”

- Lewis Carroll, Alice in Wonderland.

Questions marked with [R] are routine, [H] harder and [M] Matlab. You should make surethat you can do the easier questions before you tackle the more di!cult questions.

Problems 6.1

1. [R] Solve (if possible) the following equations for x " N, x " Z, x " Q and x " R.

a) x+ 25 = 0, 3x# 9 = 0, 3x+ 9 = 0, 3x+ 10 = 0.

b) x2 + 4x# 5 = 0, 2x2 # 13x+ 15 = 0, x2 # x# 1 = 0, x2 + 3x+ 4 = 0.

c) sin(!x/3) = 0, sin(x/3) = 0.

2. [R] Is the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} closed under addition? Prove your answer.

3. [H] Can any finite set of integers be closed under addition? Prove your answer.

4. [R] Is the set {#1, 1} closed under multiplication and division?

Problems 6.3

5. [R] Let z = 2 + 3i, w = #1 + 2i. Calculate 3z, z2, z + 2w, z(w + 3),z

w,w

z.

6. [R] Write the following expressions in a+ ib or “Cartesian” form:

a)1 + i

1 + 2i, b)

2# i

3 + i# 3# i

2 + i.

7. [R] If z = a+ ib, express the following in “Cartesian” form:

a) z2, b)1

zc)

z + 1

z # 1

8. [R] Use the quadratic formula to find all complex roots of the following polynomials.

a) z2 + z + 1, b) z2 + 2z + 3, c) z2 # 6z + 10,

d) # 2z2 + 6z # 3, e) z4 + 5z2 + 4.

9. [H] Show that!"$

3 + 1#

+"$

3# 1#

i$3

= 16(1 + i).

10. [R] Simplify"$

3 + 4i+$3# 4i

#2(where we assume

$z has non negative real part).

11. [H] Simplify

%

a+ bi

a# bi

&2

#%

a# bi

a+ bi

&2

where a and b are real numbers not both zero.

2

PROBLEMS FOR CHAPTER 6 3

Problems 6.4

12. [R] Find Re(z), Im(z) and z for z = #1 + i, 2 + 3i, 2# 3i,2# i

1 + i,

1

(1 + i)2.

13. [R] Let z = 1 + 2i and w = 3# 4i. Calculate z2 andz

w, expressing the answers in Cartesian

form.

14. [R] Given that 2z + 3w = 1 + 12i and z # w = 3# i, find z and w.

15. [R] By evaluating each side of the equations, check that zw = z w, and' z

w

(

=z

ware satisfied

by the complex numbers z = 2 + 3i, w = #1 + 2i.

16. [R] Prove that for any two complex numbers z and w

a) Im(z) =1

2i(z # z) b) 2Re(z) = z + z

c) (z # w) = z # w d)

%

1

z

&

=1

z,

e) zw = z w f)' z

w

(

=z

w.

17. [H] a) Use the properties of the complex conjugate to show that if the complex number "is a root of a quadratic equation ax2 + bx+ c = 0 with a, b, c being real coe!cients,then so is ".

b) Write down the monic quadratic polynomial with real coe!cients which has 3# 2i asone of its roots.

c) Does the result of a) generalise to higher degree polynomials?

Problems 6.5, 6.6

18. [R] Find the modulus, argument and polar form of each of the following numbers and plotthem on an Argand diagram:

a) 6 + 6i, b) # 4, c)$3# i, d)

#1$2# i$

2, e) # 7 + 3i.

19. [R] If z = 4 + 3i and w = 2 + i find |3z # 3iw|, Im"

(1# i)z # 3|w|#

.

20. [H] If z = 1 + i, calculate the powers zj for j = 1, 2, . . . , 10 and plot them on an Arganddiagram. Is there a pattern? What is the smallest positive integer n such that zn is a realnumber?

21. [R] Find the “a+ ib” form of the complex numbers whose modulus and argument are

3


a) |z| = 3, Arg(z) =!

3; b) |z| = 3, Arg(z) =

5!

6;

c) |z| = 3, Arg(z) = #2!

3; d) |z| = 3, Arg(z) = #!

6;

e) [H] |z| = 3, Arg(z) =!

8.

22. [R] a) Show that z z = |z|2. Hence, or otherwise, show that if |z| = 1, then z = z!1.

b) Show that |z| = |z| for all z " C.

c) If z = r(cos # + i sin #), show that a polar form for the complex conjugate isz = r (cos(##) + i sin(##)).

23. [H] Show that Re

%

1# z

1 + z

&

= 0 for any complex z with |z| = 1.

24. [H] Use zz = |z|2 to prove the identity |z1 + z2|2 + |z1 # z2|2 = 2(|z1|2 + |z2|2).

25. [H] Use zz = |z|2 to show that

|1# zw|2 # |z # w|2 = (1# |z|2) (1# |w|2)

and deduce that |1# zw|2 = |z # w|2 if either z or w lies on the unit circle.

Problems 6.7

26. [R] Plot the following complex numbers on an Argand diagram:

a) 2ei!

4 , b) 3e5i!

6 , c) e!2i!

3 , d) 2e!i!

2 , e) 4ei!.

27. [R] Let z = (1# i) and w = 2ei!/3. Calculate w6, z # w andw

zand express your answers in

Cartesian form.

28. [R] For z = 3e!5!i/6 and w = 1 + i, find Re"

iw + z2#

.

29. [R] Solve |ei" # 1| = 2 for #! < # ! !.

30. [R] Find Arg(#1 + i) and Arg(#$3 + i) and hence find the argument of (#1 + i)(#

$3 + i)

and the argument of#1 + i

#$3 + i

. (Make sure you get the argument in the interval (#!, !].)

31. [H] Let z = (1 +$3i) and w = (1 + i). Find Arg z and Argw and hence Arg(zw). Evaluate

zw and hence show that cos7!

12=

1#$3

2$2

. Find a similar expression for sin7!

12.

32. [R] Find polar forms for z = 1 + i$3 and w = 1# i, and hence find first the polar forms and

then the “a+ ib” forms of zw, z9, and' z

w

(12.

4


33. [R] Find the polar, and hence also the Cartesian form for:

a)"$

3 + i#5

, b)

%

#1 + i$2

&1002

, c)

)

1 +$3i

2

*!8

.

34. [H] Find the square roots (in Cartesian Form) of

a) 21# 20i, b) # 16 + 30i, c) 24 + 70i.

35. [H] a) Explain why multiplying a complex number z by ei" rotates the point represented byz anticlockwise about the origin, through an angle #.

b) The point represented by the complex number 1 + i is rotated anticlockwise about

the origin through an angle of!

6. Find its image in polar and Cartesian form.

c) Find the complex number (in Cartesian form) obtained by rotating 6#7i anticlockwise

about the origin through an angle3!

4.

36. [H] If z = rei", 0 ! # !!

2, show that

a)+

+(1# i)z2+

+ =$2r2 b) Arg

"

(1# i)z2#

= 2# # !4,

c)

+

+

+

+

+

1 + i$3

z

+

+

+

+

+

=2

r, d) Arg

)

1 + i$3

z

*

=!

3# #.

37. [R] Find the roots (in Cartesian Form) of

a) z2 # 3z + (3# i) = 0, b) z2 # (7# i)z + (14# 5i) = 0,

c) z2 + (4# i)z + (1 + 13i) = 0.

38. [R] Find the seventh roots of #1 and plot the roots on an Argand diagram.

39. [R] Find the sixth roots of i and plot the roots on an Argand diagram.

40. [R] Find the fifth roots of 16# 16i$3 and plot the roots on an Argand diagram.

41. [H] Find all z " C satisfying (z # 6 + i)3 = #27.

42. [H] Show that if $ is an nth root of unity ($ %= 1 and n > 1) then

$ + $2 + · · · + $n = 0.

Hint: Sum the geometric progression.

43. [H] Show that the set {z " C : |z| ! 1} is closed under multiplication. Is the set closed underdivision (zero excluded)? Is the set closed under addition or subtraction?

44. [H] Use the properties of complex conjugates to show that if a, b " R and |z| = 1, then|a+ bz| = |az + b|. Hint: You might find the results of Question 22 useful.

5


45. [H] Suppose a and b are real numbers (not both zero) and w =az + bz!1

bz + az!1. Show that if

|z| = 1, then |w| = 1.

HINT: You might find the results of Question 22 useful.

46. [H] Let z, w be complex numbers.

a) Using polar forms, show that |Re(zw)| ! |z| |w|.b) Use the result in (a) to show that |z + w| ! |z| + |w|, and interpret the result

geometrically. Hint: Write |z + w|2 = (z + w)(z + w) and expand.

47. [H] Let

z =i(1 + is)

1# iswhere s " R.

a) Show that

Arg(z) =

,

-

-

-

.

-

-

-

/

!

2+ 2 tan!1 s for s ! 1

#3!

2+ 2 tan!1 s for s > 1.

b) Describe geometrically what happens to z as s increases from #& to &.

48. [H] Suppose #,% %= !

2(2k + 1) where k is an integer. Use the fact that

z =1 + z

1 + z!1

a) to find the real and imaginary parts of

1 + cos 2# + i sin 2#

1 + cos 2# # i sin 2#;

b) to show that if n is a positive integer then%

1 + sin%+ i cos%

1 + sin%# i cos%

&n

= cosn'!

2# %

(

+ i sinn'!

2# %

(

.

49. [H] For n > 1, let $1,$2, ...,$n be the n distinct nth roots of 1 and let Ak be the point on theArgand diagram which represents $k. Let P represent any point z on the unit circle, andlet PAk denote the distance from P to Ak.

a) Prove that (PAk)2 = (z # $k)(z # $k).

b) Deduce thatn

0

k=1

(PAk)2 = 2n.

c) Now let P represent the point x on the real axis, #1 < x < 1, prove thatn

1

k=1

PAk = 1# xn.

6


Problems 6.8

50. [R] Using De Moivre’s theorem and the binomial theorem, prove the identity

cos 3# = 4cos3 # # 3 cos #.

51. [R] a) Use De Moivre’s Theorem to express cos 6# and sin 6# in terms cos # and sin #.

b) Write cos 6# in terms of cos # only

52. [R] Express cos 7# and sin 7# in terms of powers of cos # and sin #.

53. [R] a) Derive a formula for cos # in terms of ei" and e!i".

b) Deduce a formula for cos6 # in terms of cos k#, 1 ! k ! 6.

c) Show that

2 !

2

0cos6 # d# =

5!

32.

54. [R] Express sin5 # and cos4 # in terms of sines or cosines of multiples of #, and hence find theirintegrals.

55. [H] a) Use De Moivre’s Theorem to express cos 5# as a polynomial p(x) in x = cos #.

b) Put # = 36" = !5 and show that x = cos !

5 is a root of P (x) = 16x5 # 20x3 + 5x+ 1.

c) Check that P (x) = (x+ 1)(4x2 # 2x# 1)2.

d) What are the 5 roots of P (x)? Give full reasons for your answer.

e) Deduce that cos !5 + cos 3!

5 + cos 7!5 + cos 9!

5 = 1 and cos !5 cos

3!5 cos 7!

5 cos 9!5 = 1

16 .

56. [H] Let $1, $2, . . . ,$n be the n distinct nth roots of unity (n " 1). Show that if k is an integerthen

$k1 + $

k2 + · · ·+ $k

n

equals 0 or n. Find the values of k for which the sum is n.Hint: Write the roots in polar form and sum the resulting geometric progression. SeeExample ?? of Section 6.8.

57. [H] Show that if # is not a multiple of 2!, then the imaginary part of

1# ei(n+1)"

1# ei"is

sin"

12 (n+ 1)#

#

sin(12n#)

sin 12#

.

Hint. See Example ?? of Section 6.8.

58. [H] Find the sum ofsin # + sin 2# + · · ·+ sinn#.

Hint. See previous exercise.

7


59. [H] a) Calculate the sum of the series

S = ei" # e3i"

32+

e5i"

34# e7i"

36+ · · ·

b) Hence show that

sin # # sin 3#

32+

sin 5#

34# sin 7#

36+ · · · = 72 sin #

82 + 18 cos 2#.

Problems 6.9

60. [R] Sketch the set of points on the complex plane corresponding to each of the following:

a) |z # i| ! 2, b) |z # i| ! 2 or # !3! Arg (z # i) !

2!

3,

c) |z| " 2 and |Im(z)| ! 3, d) Re(z) " Im(z),

e) |z # i| = |z + i|, f) |z # 1# i| < 1 and # !4< Arg (z # 1# i) !

!

2,

g) |z # i| = 2|z + i|, h) [H] |z # i|+ |z + i| = 6.

61. [R] Sketch the following on two carefully labelled Argand diagrams.

a) S1 = {z : Re(z) " 3 Im(z) and |z # (3 + i)| > 2},

b) S2 =3

z : |z # i| < |z + i| and # !6! Arg (z # i) !

!

6

4

.

62. [R] Let S = {z " C : Im(z) > #4 and |z # 1# i| " 3}.

a) Sketch S on a carefully labelled Argand diagram.

b) Does 2 + 4i belong to S?

63. [R] Let z be a complex number. Prove that |z#Re(z)| ! |z# x| for all real numbers x. Drawa sketch to illustrate the result.

64. [H] Let z, w be complex numbers.

a) Sketch the subset of the complex plane defined by w = ei# for #! < " ! !.

b) Given that Arg(z) = #, prove that |z # ei"| ! |z # ei#| for all " " R.

c) Give a geometric interpretation of the result in part b).

Problems 6.10

65. [R] Use the remainder theorem to find the following remainders when.

a) 2 + 3z # z2 + 6z3 is divided by z # 5,

b) 1# 6z + 5z2 # 8z3 + 2z4 is divided by z + 2,

c) 3z + 2z2 + z3 is divided by z # 1# i.

8


66. [R] Use the remainder theorem and the factor theorem to show that z # 2 is a factor ofp(z) = 30# 17z # 3z2 + 2z3. Then divide p by z # 2 and hence find all linear factors of p.

67. [R] Use the method of the previous question to show that z # 1 and z + 2 are factors ofp(z) = #8# 6z + 7z2 + 6z3 + z4. Then find all linear factors of p.

68. [R] Find all linear factors of

a) z5 + i, b) z6 + 8.

69. [R] a) Factorise x8 # 1 into real linear and real quadratic factors.

b) Repeat for x6 + 8.

70. [R] Factorise z4 + 4 over the rational numbers.

71. [R] Factorise the polynomial z4 + i into complex linear factors.

72. [R] a) Solve the equation z6 = #1 where z " C.

b) Plot your solutions from part a) as points in the Argand diagram.

c) Write z6 + 1 as a product of complex linear factors.

d) Write z6 + 1 as a product of real quadratic factors.

73. [H] Let p(z) = z6 + z4 + z2 + 1.

a) By using the identity, (z2 # 1)p(z) = z8 # 1, find all 6 complex roots of p(z) in polarform.

b) Hence factorise p(z) into complex linear factors.

c) Factorise p(z) into a product of 3 real irreducible quadratic polynomials.

74. [H] Let p(z) = 1 + z + z2 + z3 + z4.

a) Solve z5 # 1 = 0 and hence factorise p(z) into linear factors.

b) Find all linear and quadratic factors with real coe!cients for p(z).

c) Divide the equation p(z) = 0 by z2. Let x = z + 1z and deduce that x2 + x# 1 = 0.

d) Deduce that

cos2!

5=

#1 +$5

4and cos

4!

5=

#1#$5

4

75. [H] Consider f(t) = t6 + t5 # t4 # 5t3 # 6t2 # 6t# 4. Given that #1+ i is a root of f and thatf also has two real integer roots,

a) factorise f into complex linear factors,

b) factorise f into linear and quadratic factors with real coe!cients.

9


76. [H] Let f(z) = z5 # 2z4 +2z3 # 5z2 +10z # 10. Given that 1+ i is a root, find all solutions tof(z) = 0.

77. [H] a) By considering z9 # 1 as a di"erence of two cubes, write

1 + z + z2 + z3 + z4 + z5 + z6 + z7 + z8

as a product of two real factors one of which is a quadratic.

b) Solve z9 # 1 = 0 and hence write down the six solutions of z6 + z3 + 1 = 0.

c) By letting y = z + 1z and dividing z6 + z3 + 1 = 0 by z3, deduce that

cos2!

9+ cos

4!

9+ cos

8!

9= 0.

78. [H] Let p(z) = 3z # z3 + 5z4 + z6. You are told that the six roots of p, say "1, . . . ,"6, aredistinct.

a) Prove that at least two of these roots are real.

b) Show that"1 + "2 + "3 + "4 + "5 + "6 = 0.

c) Hence or otherwise, show that there is at least one root with positive real part, andat least one root with negative real part.

d) Show that if |z| > 3, then|3z # z3 + 5z4| < |z6|.

e) Hence or otherwise show that for j = 1, . . . , 6, |"j | ! 3.

79. [H] Cardan (approximately 1545) gave a formula for the roots of the cubic equation

x3 + ax+ b = 0

in the form x = u# v, where

u =

,

.

/

# b

2+

5

%

b

2

&2

+a3

27

6

7

8

1

3

v =

,

.

/

b

2+

5

%

b

2

&2

+a3

27

6

7

8

1

3

and where the cube roots u and v must be selected to satisfy uv = a3 . It can be shown

that there are three pairs of values of u and v which satisfy the above conditions.

If a %= 0 a simpler way of writing Cardan’s formula is that the three roots of the cubic areof the form

x = u# a

3u,

where u satisfies

u3 = # b

2+

5

%

b

2

&2

+a3

27.

10


a) Use the simpler version of Cardan’s formula to find all three roots of x3# 6x+4 = 0.(Note that complex numbers are used in the calculation even though all three rootsare real).

b) Use the fact that one root of the cubic x3#6x+4 is 2 to factor the cubic as (x#2)q(x)where q(x) is a quadratic. Hence find all roots of the cubic. Hence deduce that

cos5!

12=

#1 +$3

2$2

and cos11!

12= #1 +

$3

2$2

.

80. [H] a) Show that cos 3# = 4cos3 # # 3 cos #.

b) Make the substitution x = k cos # in the equation x3 # ax# b = 0 and show that theleft hand side will become cos 3# if we choose k such that k2 = 4

3a (assume a > 0!).

c) With the above choice of k show that the cubic equation can then be written as

cos 3# =4b

k3provided # 1 !

4b

k3! 1.

d) Use the method outlined above to find the three real roots of x3 # 6x# 4 = 0.

81. [H] Consider the polynomial

p(x) = anxn + an!1x

n!1 + · · ·+ a1x+ a0

where the all coe!cients a0, a1, . . . , an are integers. Show that, if r/s is a rational root ofp for which the integers r and s have no common factors, then r is a divisor of a0 and s isa divisor of an. Hence find all rational roots of

a) #5 + 3x# x2 + 3x3, b) 5# 22x+ x2 + 28x3, c) 4# x# 100x2 + 25x3.

82. [H] Let M,N be positive integers. If xM (1# x)N is divided by (1 + x2), and the remainder is

ax+ b, show that a = ($2)N sin

(2M #N)!

4and b = (

$2)N cos

(2M #N)!

4.

Problems 6.11

83. [R] Are the solutions xn to the following discrete time systems stable (|xn| ' 0 as n ' &),or do they blow up (|xn| ' & as n ' &), or does something else happen?

a) xn+1 + xn +5

4xn!1 = 0. b) 2xn+1 + 3xn + 2xn!1 = 0.

c) xn+1 + xn + 2xn!1 = 0.

84. [R] Are the solutions x(t) to the following continuous time systems stable (|x(t)| ' 0 ast ' &), or do they blow up (|x(t)| ' & as t ' &), or does something else happen?

a)d2x

dt2+

dx

dt+ 2x = 0. b) 3

d2x

dt2# 2

dx

dt+ 2x = 0.

c)d2x

dt2+ x = 0.

11


Problems 6.12

85. [R] Prove by mathematical induction that for all positive integers n,

1.2 + 2.3 + · · · + n(n+ 1) =1

3n(n+ 1)(n + 2).

86. [R] Prove that, for all integers n " 1,

12 + 22 + 32 + · · ·+ n2 =1

6n(n+ 1)(2n + 1).

87. [R] Prove that, for all integers n " 1,

13 + 23 + 33 + · · ·+ n3 =1

4n2(n+ 1)2.

88. [H] Prove that, for all integers n " 1,

14 + 24 + 34 + · · ·+ n4 =1

30n(n+ 1)(6n3 + 9n2 + n# 1).

89. [H] Prove, by induction, that the sum to k terms of

12 # 32 + 52 # 72 + 92 # 112 + · · ·

is #8n2 if k = 2n and 8n2 + 8n+ 1 if k = 2n + 1.

90. [H] A sequence of real numbers {an}#n=1 is defined recursively by

a1 = 1, an+1 =n

0

j=1

aj2j

for n " 1.

Use the second Principle of Induction to prove that an ! 1 for all n " 1.

91. [H] Suppose we draw n lines in the plane with no three lines concurrent and no two linesparallel. Let sn denote the number of regions into which these lines divide the plane. Forexample, s1 = 2, s2 = 4, s3 = 7, . . . . Prove that sn+1 = sn + (n+ 1). Deduce by inductionthat sn = 1

2n(n+ 1) + 1.

12


Problems 6.14

92. [M] Use Matlab to evaluate (5 + i)4 (239# i), and to check numerically that

!

4= 4 cot!1 5# cot!1 239. (1)

Then use de Moivre’s Theorem to show (1).

HINT: Use format long to display more significant digits, and remember that floatingpoint calculations are done with an accuracy determined by the relative machine precisiongiven by the Matlab command eps.

93. [M] Let z = #$3 + i and w =

z

2.

a) Evaluate u = z123 by hand and using Matlab.

b) Evaluate v = w123 by hand and using Matlab.

c) Does u = 2123v in Matlab? Remember that Matlab does floating point arithmeticwhich only has around 15 decimal digits of accuracy.

“I can do Addition,” she said, “if you give me time

—- but I can’t do Subtraction under ANY circumstances!”

Lewis Carroll, Through the Looking Glass.

13


Chapter 7

VECTOR SPACES

7.1 Definitions and examples of vector spaces

7.2 Vector arithmetic

7.3 Subspaces of Rn

7.4 Linear combinations and spans

7.5 Linear independence

7.6 Basis and dimension

7.7 Coordinate vectors

7.8 Further important examples of vector spaces

7.9 Data fitting and polynomial interpolation

7.10 Appendix: A brief review of set and function notation

7.11 Vector spaces and Matlab

15

16 CHAPTER 7. VECTOR SPACES


You should try to solve some of the questions in Sections 7.4 to 7.8 with Matlab.

Problems 7.1

1. [R] Show that the setS =

9

x " R3 : x1 ! 0, x2 " 0:

,

with the usual rules for addition and multiplication by a scalar in R3 is not a vector spaceby showing that at least one of the vector-space axioms is not satisfied. Give a geometricinterpretation of this result.

2. [R] Show that the system S with the usual rules for addition and multiplication by a scalarin R3, and where

S =9

x " R3 : 2x1 + 3x32 # 4x23 = 0:

,

is not a vector space by showing that at least one of the vector-space axioms is not satisfied.

3. [R] Let S =

,

.

/

;

<

abc

=

> " R3 : (a# b)c = 0

6

7

8

.

a) Write down two non-zero elements of S.

b) Show that S is not closed under vector addition.

4. [H] The set Cn is a vector space over C (see Example ?? of Section 7.1). Check that axioms1, 2, 6, 9 are satisfied by this system.

5. [H] Let Mmn(C) be the set of all m ! n matrices with complex entries with addition theusual rule for addition of complex matrices, and multiplication by a scalar the usual rulefor multiplication of a complex matrix by a complex scalar. Prove that the vector spaceMmn(C) satisfies axioms 1, 3, 6 and 10.

6. [H] Prove that the system (Cn,+, (,R) with “natural” definitions of + and ( is a vector space,whereas the system (Rn,+, (,C) with “natural” definitions of + and ( is not a vectorspace.

7. [H] Consider the system (R2,),*,R) in which the usual operations of “addition” and “mul-tiplication by a scalar” are replaced by the new definitions:

%

a1a2

&

)%

b1b2

&

=

%

a1 + b1a2 # 3b2

&

&*%

a1a2

&

=

%

4&a1&a2

&

.

Give a list of the vector-space axioms satisfied by this system, and a list of any which arenot satisfied. Is this system a vector space?

16


Problems 7.2

8. [H] Prove that the following properties are true for every vector space.

a) 2v = v + v.

b) nv = v + · · ·+ v, where there are n terms on the right.

9. [H] Prove parts 2, 4 and 5 of Proposition ?? of Section 7.2.

10. [R] Let A, B, P be points in R3 with position vectors

a =

;

<

7#23

=

>, b =

;

<

1#50

=

> and p =

;

<

1#12

=

>.

Let Q be the point on AB such that AQ = 23AB.

a) Find q, the position vector of Q.

b) Find the parametric vector equation of the line that passes through P and Q.

11. [R] Consider the three points A(1, 1, 1), B(2, 0, 3) and C(3,#1, 1).

a) Find##'AB and

#'AC.

b) Find a parametric vector form of the line through A and B.

c) Find a parametric vector form of the plane through A, B and C.

d) Find##'AB !#'

AC.

e) Find a point-normal form of the plane through A, B and C.

f) Find a Cartesian equation of the plane through A, B and C.

12. [R] Given the vectors p =

;

<

11#1

=

> and q =

;

<

21#1

=

>, find |p|, |q|, p · q, then the cosine of the

angle between p and q.

13. [R] Consider the equation

det

;

<

x# 1 y # 2 z + 11 0 22 #1 0

=

> = 0

a) Show that the equation represents the Cartesian equation of a plane.

b) Write the equation in point-normal form.

14. [R] For the points P (1, 2, 0) , Q(1, 3,#1) and R(2, 1, 1), find##'PQ ! #'

PR and the area of thetriangle with vertices P, Q and R.

17


15. [R] Suppose that A is the point (2,#1, 3) and # is the plane

x = &

;

<

101

=

> + µ

;

<

1#22

=

> for &, µ " R.

for &, µ " R.

a) Find a vector n which is normal to #.

b) Find the projection of#'OA on the direction n.

c) Hence find the shortest distance of A from #.

Problems 7.3

16. [R] Suppose that v is a vector in Rn. Show that the line segment defined by

S = {x " Rn : x = &v, for 0 ! & ! 10}

is not a subspace of Rn.


9

x " R3 : 2x1 + 3x2 # 4x3 = 6:

,

is not a subspace of R3. Give a geometric interpretation of this result.

18. [R] Let S the setS =

9

x " R3 : 2x1 + 3x2 # 4x3 = 0:

,

a) Find three distinct members of S.

b) Show that S is a subspace of R3.

c) Give a geometric interpretation of this latter result.

19. [R] Show that

T =

,

.

/

;

<

xyz

=

> : #1 ! x+ y + z ! 1

6

7

8

is not a vector subspace of R3.

20. [R] Show that the set

S =9

x " R3 : 2x1 + 3x2 # 4x3 = 4x1 # 2x2 + 3x3 = 0:

is a subspace of R3.


S =9

x " R3 : 2x1 + 3x2 # 4x3 = 0 or 4x1 # 2x2 + 3x3 = 0:

is not a subspace of R3.

18



S =9

b " R2 : b = Ax for some x " R3:

,

where

A =

%

2 #3 14 5 #3

&

,

is a subspace of R2. Explain why each column of the matrix belongs to the set S.

23. [R] For each of the following subsets of R3, either prove that the given subset is a subspace ofR3 or explain why it is not a subspace.

a) S =

,

.

/

;

<

x10x3

=

> " R3 : x1 + 2x3 " 0

6

7

8

.

b) T =

?

40

i=1

&ivi : &i " R, 1 ! i ! 4

@

, where v1, v2, v3, v4 are given fixed vectors in

R3.

c) U =9

Ax : x " R5:

, where A is a fixed 3! 5 matrix.

24. [H] Suppose that u =

;

<

10#2

=

> and v =

;

<

123

=

>. Show, by the Subspace Theorem that the set

S =9

x " R3 : x = &u+ µv, for &, µ " R:


25. [H] Prove that the set S in Example ?? on page ?? is closed under multiplication by a scalar.

26. [H] Let a and b be two fixed non-zero vectors in R5. Show that

W =9

x " R5 : x · a = x · b = 0:


If a = e1 =

;

A

A

A

A

<

10000

=

B

B

B

B

>

and b = e2 =

;

A

A

A

A

<

01000

=

B

B

B

B

>

, describe W.

27. [R] Show that the set S = {p " P2 : p(0) = 1} is NOT a subspace of P2.


9

p " P3 : p$$(x) = 0 for all x " R:

is a subspace of P3.

19


29. [R] Is the setS =

9

p " P3 : p$(x) + x+ 1 = 0 for all x " R:

a subspace of P3?

30. [R] Consider the set

S =9

p " P3 : (x+ 1)p$(x)# 3p(x) = 0 for all x " R:

.

a) Show that S is a subspace of P3, (the set of all real polynomials of degree ! 3).

b) Find a polynomial in S where not all the coe!cients are zero.

31. [H] By constructing a counterexample, show that the union of two subspaces is not, in general,a subspace.

32. [H] Let W1 and W2 be two subspaces of a vector space V over the field F. Prove that theintersection of W1 and W2 (i.e., the set W1 +W2) is a subspace of V .

33. [H] Let V be a vector space over the field F.

a) Let {Wk : 1 ! k ! m} be m subspaces of V , and let W be the intersection of thesem subspaces. Prove that W is a subspace of V .

b) Let S be any set of vectors in V , and let W be the intersection of all subspaces of Vwhich contain S (that is, x " W if and only if x lies in every subspace which containsS). Prove that W is the set of finite linear combinations of vectors from S.

Problems 7.4

34. [R] Let a =

;

<

10114

=

> , v1 =

;

<

214

=

> , v2 =

;

<

#1#21

=

> and v3 =

;

<

33#1

=

> .

a) Does a " span(v1,v2,v3)? If so, express a as a linear combination of v1,v2 and v3.

b) Do the vectors v1,v2,v3 span R3? If not, find condition(s) on b =

;

<

b1b2b3

=

> for b to

belong to span(v1,v2,v3) and interpret your answer geometrically.

35. [R] Repeat the preceding question using

a =

;

<

9#2#4

=

> , v1 =

;

<

023

=

> v2 =

;

<

5#2#3

=

> and v3 =

;

<

15#4#6

=

>.

36. [R] Repeat using a =

;

A

A

<

11#91

=

B

B

>

, v1 =

;

A

A

<

1305

=

B

B

>

, v2 =

;

A

A

<

2212

=

B

B

>

, v3 =

;

A

A

<

#1041

=

B

B

>

and b =

;

A

A

<

b1b2b3b4

=

B

B

>

. [Replace

R3 by R4, of course.]

20


37. [R] Is the vector b =

;

<

#2#6#4

=

> " span (v1,v2,v3,v4), where

v1 =

;

<

130

=

>, v2 =

;

<

221

=

>, v3 =

;

<

#10#1

=

>, v4 =

;

<

1#21

=

>?

38. [R] Is the set of vectors v1 =

;

<

123

=

>, v2 =

;

<

11#1

=

>, v3 =

;

<

#105

=

> a spanning set for R3?

39. [R] Does v belong to the column space of A, col(A), where

v =

;

A

A

<

2#519

#13

=

B

B

>

and A =

;

A

A

<

1 3 #10 1 22 #3 #51 2 7

=

B

B

>

?

If so, write v as a linear combination of the columns of A.

40. [R] Is the polynomial p(x) = 1 + x+ x2 in span"

1# x+ 2x2, #1 + x2, #2# x+ 5x2#

?

41. [R] Is S =9

1 + x, 1# x2, x+ 2x2:

a spanning set for P2?

42. [H] Prove Proposition ?? of Section 7.4.

43. [H] Use the vector-space axioms to prove that we do not need to use brackets when writingdown the linear combination

n0

k=1

&kvk = &1v1 + · · ·+ &nvn.

That is, prove that the result of the operations in independent of the order in which theadditions are performed.

Problems 7.5

44. [R] Is the set of vectors v1 =

;

<

123

=

>, v2 =

;

<

11#1

=

>, v3 =

;

<

#105

=

> linearly independent? Are

these three vectors coplanar?

45. [R] Is the set S = {v1,v2,v3}, where v1 =

;

<

11#1

=

>, v2 =

;

<

2#10

=

>, v3 =

;

<

5#43

=

>, a linearly

independent set? Are these three vectors coplanar?

46. [R] Can a set of linearly independent vectors contain a zero vector? Explain your answer.

21


47. [R] Given the set S = {v1,v2,v3}, where v1 =

;

<

1#3#2

=

>, v2 =

;

<

321

=

>, v3 =

;

<

4#1#1

=

>, do the

following.

a) Show that S is a linearly dependent set.

b) Show that at least one of the vectors in S can be written as a linear combination ofthe others, and find the corresponding linear combination.

c) Find all possible ways of writing the vector

;

<

895

=

> as a linear combination of the

vectors in the set.

d) Find a linearly independent subset of S with the same span as S, and then show that;

<

895

=

> can be written as a unique linear combination of this subset.

e) Give a geometric interpretation of span (S).

48. [R] Repeat the previous question for the set of four vectors S = {v1,v2,v3,v4}, where

v1 =

;

<

130

=

>, v2 =

;

<

221

=

>, v3 =

;

<

#10#1

=

>, v4 =

;

<

1#21

=

>.

49. [R] Is9

1# x+ 2x2, #1 + x2, #2# x+ 5x2:

a linearly independent subset of P2? If the setis not linearly independent express one of the polynomials as a linear combination of theothers.

50. [H] (For discussion). Let the set S = {v1, . . . ,vn} be a linearly independent subset of a vectorspace V . You are standing at the origin of V and set o" in the direction of v1. After acertain length of time, you turn and head in direction v2 — then in direction v3 and soon. Is it possible for you to return to the origin? (Note: You may walk any distance thatyou like along any of the directions, but you are not allowed to retrace your steps).

51. [H] What would happen in the previous question if the set S were a linearly dependent set?

52. [H] Assume that m ! n and that S = {v1, . . . ,vm} is a set of mutually orthogonal, non-zerovectors in Rn, that is, the dot products satisfy (see Section 5.3.1)

vi · vj = 0 for i %= j; 1 ! i, j ! m

vi · vi %= 0 for 1 ! i ! m.

Show that S is a linearly independent set.

22


Problems 7.6

53. [R] Is the set S = {v1,v2,v3}, where v1 =

;

<

11#1

=

>, v2 =

;

<

2#10

=

>, v3 =

;

<

5#43

=

>, a basis for R3?

54. [R] Find a basis for, and the dimension of, W = span (v1,v2,v3), where

v1 =

;

<

123

=

>, v2 =

;

<

11#1

=

>, v3 =

;

<

#105

=

>.

55. [R] Without doing any calculation, explain why

,

-

-

.

-

-

/

;

A

A

<

1305

=

B

B

>

,

;

A

A

<

2213

=

B

B

>

,

;

A

A

<

#1040

=

B

B

>

6

-

-

7

-

-

8

is not a spanning

set for R4.

Similarly, without doing any calculation, explain why

,

.

/

;

<

130

=

> ,

;

<

221

=

> ,

;

<

#101

=

> ,

;

<

1#31

=

>

6

7

8

is a linearly dependent set.

56. [R] Which of the following statements are true and which are false? Explain your answer.

a) Any set of 6 vectors in R5 is linearly dependent.

b) Some sets of 6 vectors in R5 are linearly independent.

c) Any set of 6 vectors in R5 is a spanning set for R5.

d) Some sets of 6 vectors in R5 span R5.

e) Same as in (a) – (d), with 6 replaced by 4.

f) Any set of 5 vectors in R5 is a basis for R5.

g) Some sets of 5 vectors in R5 are bases for R5.

h) Any set of vectors which spans R5 is linearly independent.

i) Any set of 5 vectors which spans R5 is linearly independent.

j) Any 5 linearly independent vectors in R5 form a basis for R5.

57. [R] Let V be a finite dimensional real vector space, and let S = {v1, . . . ,vn} be a finite setof vectors in V . Suppose also that the dimension of V is '. State, with brief reasons, therelationship, if any, between n and ' if

a) S is linearly independent.

b) S is linearly dependent.

c) S spans V .

d) S is a basis for V .

58. [H] Explain why it is impossible to have a set of m mutually orthogonal, non-zero vectors inRn with m > n.

23


59. [R] Consider the plane P in R3 whose equation is

x+ y + z = 0.

a) Prove that P is a subspace of R3.

b) Find a basis for P . Give reasons for your answer.

60. [R] Find a basis for, and the dimension of, the column space of the matrix

A =

;

A

A

<

1 1 #1 #2 10 0 1 4 #10 0 0 2 20 0 0 0 0

=

B

B

>

.

61. [R] Find a basis for, and the dimension of, col(A), where

A =

;

A

A

<

1 1 0 2 10 0 #1 #2 2

#1 #1 1 4 #11 1 0 4 2

=

B

B

>

.

62. [R] Show that the columns of the matrix A given below are not a spanning set for R4. Thenfind a basis for R4 which contains as many of the columns of A as possible.

A =

;

A

A

<

1 3 3 #7 52 6 5 #8 13 9 5 #3 #24 12 5 2 #5

=

B

B

>

.

63. [R] Consider the set T = {v1, v2, v3, v4, x} where

v1 =

;

A

A

<

12

#10

=

B

B

>

, v2 =

;

A

A

<

36

#30

=

B

B

>

, v3 =

;

A

A

<

21

#14

=

B

B

>

, v4 =

;

A

A

<

#1#524

=

B

B

>

, x =

;

A

A

<

6#3#120

=

B

B

>

.

a) Find a basis B for span (v1, v2, v3, v4, x).

b) Explain why x belongs to span (v1, v2, v3, v4). Write x as a linear combination ofB.

c) Suppose the matrix A has columns v1, v2, v3, v4. What is the dimension of thecolumn space of A?

64. [R] Show that the set9

1# x2 + x3, x+ 2x2, 2 + x# x2 + 2x3, 2x# x2 + x3:

forms a basisfor P3.

65. [R] Consider the set of polynomials S = {p1, p2, p3, p4} in P2, where

p1(z) = 1 + z # z2, p2(z) = 2# z, p3(z) = 5# 4z + z2, p4(z) = z2.

Show that S is a linearly dependent spanning set for P2, and then find a subset of S whichis a basis for P2.

24


66. [H] You are given that V is a vector space and that S = {v1,v2,v3} is a subset of V . Supposethat w " span(S). Prove that the set

{v1,v2,v3,w}

is a linearly dependent set.

67. [H] Prove that the only subspaces of R4 are

i) {0}, where 0 is the zero vector,

ii) lines through the origin,

iii) planes through the origin,

iv) subspaces of the form span (S), where S is any set of three linearly independentvectors in R4, and

v) R4 itself.

Problems 7.7

68. [R] Show that the columns of the matrix

A =

;

A

A

<

1 2 #1 13 2 0 #20 1 #1 15 3 0 #1

=

B

B

>

are a basis for R4. Then find the coordinate vector of v =

;

A

A

<

#2#6#4#2

=

B

B

>

with respect to the

ordered basis given by the columns of A.

69. [R] A vector v " R4 has the coordinate vector

;

A

A

<

16#14

=

B

B

>

with respect to the ordered basis formed

by the columns of the matrix A of the previous question. Find v.

70. [R] Find the vector v that has coordinate vector

;

<

2#11

=

> with respect to the ordered basis

,

.

/

;

<

12#2

=

> ,

;

<

37#5

=

> ,

;

<

249

=

>

6

7

8

of R3.

71. [R] Find the coordinates of the following vectors with respect to the given ordered bases.

25


a)

;

<

123

=

> with respect to

,

.

/

;

<

101

=

> ,

;

<

111

=

> ,

;

<

#20#1

=

>

6

7

8

.

b)

;

<

a1a2a3

=

> with respect to

,

.

/

;

<

101

=

> ,

;

<

111

=

> ,

;

<

#20#1

=

>

6

7

8

.

72. [R] With respect to the basis B =

,

.

/

;

<

1#12

=

> ,

;

<

346

=

> ,

;

<

#23#3

=

>

6

7

8

of R3,

a) find the vector v with coordinate vector [v]B =

;

<

31#3

=

>;

b) find the coordinate vector of w =

;

<

7#311

=

>.

73. [H] Consider the set S = {v1,v2,v3}, where v1 =

;

A

<

1%2

# 1%2

0

=

B

>, v2 =

;

A

<

1%31%31%3

=

B

>, v3 =

;

A

<

# 1%6

# 1%6

2%6

=

B

>.

Without solving systems of linear equations, do the following.

a) Show that S is an orthonormal set of vectors in R3.

b) Show that S is a basis for R3.

c) Find the coordinate vector of

;

<

#134

=

> with respect to the ordered basis S.

HINT. See Example ?? of Section 7.6.

74. [H] Let S = {u1, . . . ,un} be an orthonormal set of n vectors in Rn. Prove that S is a basisfor Rn, and then show that the coordinate vector for any v " Rn is given by

[v]S =

;

A

<

x1...xn

=

B

>, where xj = uj · v.

Problems 7.8

75. [R] Let M22 be the vector space of all 2 ! 2 matrices with real entries (see Example ?? ofSection 7.1). Let S be the set

S = {A " M22 : a11 + a22 = 5} .

26


a) Find three matrices in S.

b) Is S a subspace of M22? Give a reason.

76. [R] Let T be the setT = {A " M22 : a11 + a22 = 0} .

a) Find three matrices in T .

b) Is T a subspace of M22? Give a reason.

77. [R] Show that the four matrices

%

1 00 0

&

,

%

0 10 0

&

,

%

0 01 0

&

,

%

0 00 1

&

.

form a basis for M22(R). This set is called the standard basis for M22(R).

78. [H] Show that the four matrices of the previous question also form a basis for the vector spaceM22(C) of all 2! 2 matrices with complex entries.HINT: Can you see why your proof of the previous question will also be valid for complexnumbers?

79. [H] Show that the set of four matrices

%

1 00 1

&

,

%

0 11 0

&

,

%

0 #ii 0

&

,

%

1 00 #1

&

.

form a basis for M22(C). These matrices are called the Pauli spin matrices, and they areimportant in quantum physics and chemistry.

80. [H] Find the coordinates of the following vectors with respect to the given ordered bases.

a) The matrix

A =

%

a11 a12a21 a22

&

with respect to the standard basis for M22 given in Question 77. Note that the resultsare the same for both real and complex numbers

b) Repeat part (b) for the basis of Pauli spin matrices given in Question 79. In this casethe entries of A should be regarded as complex numbers.

81. [R] Let R =

C%

1 0#2 0

&

,

%

0 13 0

&

,

%

0 05 1

&D

.

a) Express

%

#4 2#1 #3

&

as a linear combination of elements of R.

b) Does R span M22(R), the space of all 2 ! 2 matrices? Give a brief reason for youranswer.

27


82. [H] Complete the proof of Proposition ?? of Section 7.8 that the system (R[X],+, (,R) is avector space.

83. [H] Let C[X] be the set of all complex-valued functions with domain X. Show that the system(C[X],+, (,C), where + and ( are the usual rules for addition and multiplication by ascalar of functions, satisfies vector-space axioms 2, 4, 7 and 9. This system is a vectorspace over the complex field C.

84. [H] Show that the set

S =

C

y " R[R] :d2y

dx2+ 3

dy

dx+ 4y = 0

D

is a subspace of the vector space R[R] of all real-valued functions with domain R.

85. [H] Is the set

S =

C

y " R[R] :d2y

dx2+ 3

dy

dx+ 4y = 5

D

a subspace of R[R]? Prove your answer.

86. [H] Let C(k)[R] be the set of all real-valued functions with domain R for which the first kderivatives exist and are continuous. Prove that C(k)[R] is a subspace of the vector spaceR[R] of all real-valued functions with domain R.

87. [H] Show that the vector spaces C(k)[R] defined in the previous question have the propertythat, if m " n, then C(m)[R] is a subspace of C(n)[R].

88. [H] Let S be the subset of R [[#!,!]] defined by

S =

C

f " R [[#!,!]] :2 !

!!cos(x+ t)f(t)dt = 0 for all x " [#!,!]

D

.

Prove that S is a subspace of the vector space R [[#!,!]].

89. [H] This question generalises the results of Question 52 to real-valued functions.

Let S = {f1, . . . , fn} be a set of real-valued functions defined on an interval [a, b] with theproperties that

2 b

afi(x)fj(x)dx = 0 for i %= j; 1 ! i, j ! n

2 b

af2i (x)dx %= 0 for 1 ! i ! n.

Prove that S is a linearly independent set.

Note. A set of functions with these properties is said to be mutually orthogonal on theinterval [a, b].

28


90. [H] Show that the set

S =9

p " Pn(R) : 5p$(6) + 3p(6) = 0

:

, where p$(x) =dp

dx,

is a subspace of the vector space Pn(R) of all real polynomials of degree less than or equalto n.

91. [H] Is the set

S =9

p " Pn(R) : 5p$(6) + 3p(6) = 8

:

, where p$(x) =dp

dx,

a subspace of Pn(R)? Prove your answer.

92. [H] Let P be the set of all polynomials over the complex-number field C. Show that P is asubspace of the vector space C[C] of all complex-valued functions with domain C.

93. [R] Is the polynomial p " span (p1, p2, p3), where the polynomials are defined by

p(z) = #6+2z+30z2, p1(z) = 1+2z+3z2, p2(z) = #4#z+9z2, p3(z) = #5#z+12z2.

94. [R] Find conditions on the coe!cients of the polynomial p " P2 for p to be a linear combinationof the three polynomials p1,p2, p3, where the polynomials are given by

p1(z) = 2z + 3z2, p2(z) = 5# 2z # 3z2, p3(z) = 15# 4z # 6z2.

95. [R] Are the polynomials p1, p2, p3 in the previous two questions spanning sets for P2?

96. [R] Is the set of polynomials S = {p1, p2, p3} in P2, where

p1(z) = 1 + z # z2, p2(z) = 2# z, p3(z) = 5# 4z + z2,

a linearly independent set? If not, express one of the polynomials as a linear combinationof the others.

97. [R] Show that

p1(z) = #2 + 5z # 4z2 + 15z3 # 5z4 + z5,

p2(z) = 3z + 4z2 # 3z3 + 6z5,

p3(z) = 2 + 3z2 # 4z3 + 10z4 # 5z5,

p4(z) = 3 + 14z2 # 5z3 + 6z4 # 3z5,

p5(z) = 3 + 8z + 17z2 + 3z3 + 11z4 # z5,

p6(z) = #3 + 11z # 7z2 + 10z3 # z4 + 11z5,

are not a spanning set for P5, and then construct a basis for P5 containing as many of thegiven polynomials as possible.

HINT. Check using Matlab.

29


98. [R] Find the coordinate vector for p(x) = 1 + 2x + x2 with respect to the ordered basis9

1 + x, 1# x2, x+ 2x2:

of P2.

99. [R] Find the coordinate vector of 1+ 2z+3z2 with respect to the ordered basis of P2 given by

C

1

8z(z # 2), 1 # 1

4z2,

1

8z(z + 2)

D

.

Note. This question and the one that follows do not require Gaussian Elimination.

100. [H] Find the coordinate vector of a0+a1z+a2z2 with respect to the ordered basis of P2 givenby

C

1

2z(z # 1), 1 # z2,

1

2z(z + 1)

D

.

101. [H] Let S = {p1, . . . , pn} be a set of n polynomials in Pn!1(R) with the property that

2 b

api(x)pj(x)dx =

?

0 for i %= j

1 for i = jfor 1 ! i, j ! n.

A set of polynomials with this property is called an orthonormal set of polynomials on theinterval [a, b].

Prove that S is a basis for Pn!1(R), and then show that the coordinate vector for anyp " Pn!1(R) is given by

[p]S =

;

A

<

x1...xn

=

B

>, where xi =

2 b

api(x)p(x)dx.

Problems 7.9

102. [R] For the data in Table ?? on page ??

a) find the interpolating polynomial of degree 3 using the first 4 data points and theLagrange basis. Use this to predict the interest rate at 1 year. Is this interpolationor extrapolation? Does it agree with the data value given for 1 year?

b) Use the Matlab script polint.m available from the MATH1251 web page to fit apolynomial of degree 4 using all 5 data points and predict the rate for 2 years.

103. [H] Let the m " n points ti for i = 1, . . . ,m be distinct, and define the matrix A " Mmn byAij = %j(ti) for i = 1, . . . ,m and j = 1, . . . , n.

a) Show that if A has linearly independent columns then the functions %j(t) for j =1, . . . , n are linearly independent.

30


b) Give an example of two distinct points and two linearly independent functions forwhich the matrix A does not have linearly independent columns. Hint: Try low orderpolynomials at 0 and 1.

104. [H] a) Show that the Vandermonde determinant

+

+

+

+

+

+

+

+

+

1 t1 . . . tn!11

1 t2 . . . tn!12

...... . . .

...1 tn . . . tn!1

n

+

+

+

+

+

+

+

+

+

=1

i<j

(tj # ti)

for n = 2, 3, 4. (This formula holds for all integers n > 1).

b) What is the significance of this result for polynomial interpolation at the pointst1, t2, . . . , tn?

31


Chapter 8

LINEAR TRANSFORMATIONS

8.1 Introduction to linear maps

8.2 Linear maps from Rn to Rm and m! n matrices

8.3 Geometric examples of linear transformations

8.4 Subspaces associated with linear maps

8.5 Further applications and examples of linear maps

8.6 Representation of linear maps by matrices

8.7 Matrix arithmetic and linear maps

8.8 One-to-one, onto and invertible linear maps and matrices

8.9 Proof of the Rank-Nullity Theorem

8.10 One-to-one, onto and inverses for functions

8.11 Linear transformations and Matlab

33

34 CHAPTER 8. LINEAR TRANSFORMATIONS


Problems 8.1

1. [R] Explain why the function S : [#1, 1] ' R defined by S(x) = 5x for x " [#1, 1] is not alinear map. Then show that the function T : R ' R defined by T (x) = 5x for x " R is alinear map.

2. [R] For the following examples, determine whether T is a linear map by using Definition ??.

a) T : R2 ' R4 defined by

T (x) =

;

A

A

<

3x1 # x22x1 + 4x2#3x1 # 3x2

x2

=

B

B

>

for x =

%

x1x2

&

" R2.

b) T : R4 ' R3 defined by

T (x) =

;

<

#2x1 + 5x36x1 # 8x2 + 2x4#2x1 + 4x2 # 3x3

=

> for x =

;

A

A

<

x1x2x3x4

=

B

B

>

" R4.

c) T : R3 ' R2 defined by

T (x) =

%

3x1 + 4#2x1 + 3x2 # x3

&

for x =

;

<

x1x2x3

=

> " R3.

d) T : R4 ' R3 defined for x =

;

A

A

<

x1x2x3x4

=

B

B

>

" R4 by

T (x) = x1

;

<

13#2

=

> + x2

;

<

0#42

=

> + x3

;

<

#2#34

=

> + x4

;

<

#410

=

> .

e) T : R3 ' R2 defined by

T (x) =

%

3x22 # x3x1 # 4x2

&

for x =

;

<

x1x2x3

=

> " R3.

34


3. [H] Consider the complex numbers as a real vector space. Specify the “natural” domain andcodomain for each of the following functions of a complex number and determine if thefunction is a linear function with F = R.

a) T (z) = Re(z) , b) T (z) = Im(z), c) T (z) = |z|,d) T (z) = Arg (z) , e) T (z) = z .

4. [R] Show that the sine function T : R ' R, defined by

T (x) = sin(x) for x " R,

satisfies parts 1 and 2 of Proposition ?? of Section 8.1 but that it is not a linear map.

5. [H] Use proof by induction to prove Theorem ?? of Section 8.1.

6. [R] If {v1,v2} are linearly independent in a real vector space V and v3 = 2v1 + v2, is there alinear map T : W ' R2 where W = span(v1,v2) such that

T (v1) =

%

12

&

, T (v2) =

%

#32

&

, T (v3) =

%

#13

&

?

7. [R] A linear map T : R3 ' R4 has function values given by

T

;

<

100

=

> =

;

A

A

<

1234

=

B

B

>

, T

;

<

010

=

> =

;

A

A

<

#3014

=

B

B

>

, T

;

<

001

=

> =

;

A

A

<

40#56

=

B

B

>

. Find T

;

<

2#14

=

> and T

;

<

x1x2x3

=

>.

8. [R] Show that any function T : R3 ' R4 with function values given by T

;

<

100

=

> =

;

A

A

<

1234

=

B

B

>

,

T

;

<

010

=

> =

;

A

A

<

#3014

=

B

B

>

, T

;

<

001

=

> =

;

A

A

<

40#56

=

B

B

>

, and T

;

<

1#31

=

> =

;

A

A

<

2431

=

B

B

>

is not a linear map.

9. [R] A linear function T : R3 ' R2 has function values given by

T

;

<

123

=

> =

%

41

&

, T

;

<

#21#4

=

> =

%

#12

&

, T

;

<

1#12

=

> =

%

#4#2

&

.

Write

;

<

021

=

> as a linear combination of the vectors in the basis

,

.

/

;

<

123

=

> ,

;

<

#21#4

=

> ,

;

<

1#12

=

>

6

7

8

of R3. Hence find T

;

<

021

=

>.

HINT. Use Theorem ?? of Section 8.1.

35


10. [H] Given that T

;

<

123

=

> =

%

41

&

, T

;

<

#214

=

> =

%

#12

&

and T

;

<

1713

=

> =

%

4#2

&

, show that T is not

a linear map.

Problems 8.2

11. [R] For any function in Question 2 which is a linear map, find a matrix A such that T (x) =Ax for x in the domain by using the results of the Matrix Representation Theorem ofSection 8.2.

12. [R] For any function in Question 2 which is a linear map, write a system of linear equationsfor T (x) for x in the domain, and hence find a matrix A such that T (x) = Ax. Check thatthe matrices you obtain in this question are the same as the matrices that you obtainedin the previous question.

Problems 8.3

13. [R] For each of the following 2 ! 2 matrices, draw a picture to show Ae1, Ae2, Ab, where e1and e2 are the standard basis vectors in R2 and where b = 2e1 + 3e2.

a)

%

2 00 0.7

&

, b)

%

#2 00 2

&

, c)

%

#2 00 #3

&

, d)

%

6 #26 #1

&

, e)

%

4 #43 #4

&

.

14. [R] Draw the image of the star in Figure ??(a) on page ?? under each of the transformationsdefined by the matrices in Quesion 13.

15. [R] Let T be the rotation in the plane R2 through angle!

3in the anti-clockwise direction.

Find the matrix which represents the linear transformation T .

16. [H] Let x be the position vector of a point X in R2, and let x$ be the position vector of thepoint X $ which is the reflection of X in the x2-axis. (That is, assume that a mirror isplaced along the x2-axis and that X $ is the reflection of X in the mirror.) Show that thefunction T : R2 ' R2 defined by T (x) = x$ is a linear map. Find a matrix A which

transforms x =

%

x1x2

&

into x$ =

%

x$1x$2

&

.

17. [H] Let x be the position vector of a point X in R3 and let x$ be the position vector of the pointX $ which is the reflection of X in the (x1, x2)-plane. Show that the function T : R3 ' R3

defined by T (x) = x$ is a linear map. Find a matrix A which transforms the position

vector

;

<

x1x2x3

=

> of X into the position vector

;

<

x$1x$2x$3

=

> of X $.

36


18. [H] Let p be the position vector of a point P in Rn and let q be the position vector of thepoint Q which is the reflection of P in the line

x = &d; & " R.

Show that the function T : Rn ' Rn defined by T (p) = q is a linear map. Find a matrix

A which transforms p =

;

A

<

p1...pn

=

B

>into q =

;

A

<

q1...qn

=

B

>.

19. [R] Let b be a fixed vector in R3. Is the function T : R3 ' R3 defined by

T (x) = b! x for x " R3,

where b ! x is the cross product, a linear map? Prove your answer. Find a matrix A

which transforms the vector x =

;

<

x1x2x3

=

> into its function value T (x) =

;

<

x$1x$2x$3

=

>.

20. [H] In Example ?? of Section 8.3 we have stated that if b is a fixed non-zero vector in Rn thenthe projection function T : Rn ' Rn of vectors onto b, which is defined by

T (a) = projb(a) =a · b|b|2

b for a " Rn,

is a linear map. Prove this statement. If n = 3 and b =

;

<

102

=

>, find a matrix A which

transforms the vector a into its projection T (a).

21. [H] If a is regarded as a fixed non-zero vector, is the function S : Rn ' Rn defined by

S(b) =

?

projba for b " Rn\{0}0 b = 0

a linear map? Prove your answer.

22. [H] Let A$, A" and A$+" be the matrices for rotations in the plane by angles %, # and %+ #respectively (see Example ?? of Section 8.3). Prove that

A"A$ = A$+".

What is this saying geometrically?

23. [H] Let B = {i, j,k} be an ordered orthonormal basis (Cartesian coordinate system) for athree-dimensional geometric vector space. Let a be a three-dimensional geometric vectorand let a$ = R#(a) be the vector obtained by rotating a anticlockwise by an angle "

37


about an axis parallel to j. If the coordinate vectors of a and a$ are [a]B =

;

<

a1a2a3

=

> and

[a$]B =

;

<

a$1a$2a$3

=

>, find the rule R#

;

<

a1a2a3

=

> =

;

<

a$1a$2a$3

=

> and show that it defines a linear map from

R3 to R3. Also find the matrix A such that a$ = Aa.

Problems 8.4


,

.

/

;

<

&#2&&

=

> : & " R

6

7

8

is the kernel of the matrix

%

3 1 #18 3 #2

&

.

25. [R] Find the kernel and the nullity of each of the following matrices.

a) A =

;

<

2 #1 31 #2 34 1 #1

=

>, b) B =

;

<

0 5 152 #2 #43 #3 #6

=

>,

c) C =

;

<

1 2 #1 13 2 0 #20 1 #1 1

=

>.

Where possible, give a geometric interpretation of the kernels.

26. [R] Find a basis for the kernel, and the nullity, of each of the following matrices.

a) D =

;

<

1 2 5 01 #1 #4 0

#1 0 1 1

=

>, b) E =

;

A

A

<

1 1 #12 #1 05 #4 10 0 1

=

B

B

>

.

27. [H] Let W =3

"

x1 x2 x3 x4#T

: x1 + x2 + x3 + x4 = x1 + 2x2 + 3x3 + 4x4 = 04

. Find a

matrix A such that W = kerA.

28. [R] Find ker(T ) and nullity(T ) for the linear functions of Question 2.

29. [R] Find ker(T ) and nullity(T ) for the linear functions of Questions 16 through 20. Give ageometric interpretation of the kernels.

30. [H] Suppose that b =

;

<

123

=

>.

a) Prove that the mapping T : R3 ' R3, given by T (x) = b ! x for all x " R3, is alinear mapping.

38


b) Find the dimension of the kernel of this mapping.

31. [R] For each given vector b and matrix A, determine if b " im(A).

a) b =

;

<

11104

=

> , A =

;

<

1 #2 32 #1 34 1 #1

=

>.

b) b =

;

<

9#2#4

=

> , A =

;

<

0 5 152 #2 #43 #3 #6

=

>.

c) b =

;

<

#2#6#4

=

> , A =

;

<

1 2 #1 13 2 0 #20 1 #1 1

=

>.

32. [R] Find conditions on b1, b2, b3 for the vector b =

;

<

b1b2b3

=

> " R3 for b to belong to im(A) for

the matrices in the preceding question.

33. [R] Find a basis for the image, and the rank, of each of the matrices in Questions 25 and 26.

34. [R] By comparing the answers to Questions 25, 26 and 33, verify the conclusion of the Rank-Nullity Theorem.

35. [R] Find a basis for the image, and the rank, of each of the matrices

A =

;

A

A

<

1 1 #1 #2 10 0 1 4 #10 0 0 2 20 0 0 0 0

=

B

B

>

, B =

;

A

A

<

1 1 0 2 10 0 #1 #2 2

#1 #1 1 4 #11 1 0 4 2

=

B

B

>

.

36. [R] Find a basis for R3 which contains a basis of im(C), where

C =

;

<

1 2 3 42 #4 6 #2

#1 2 #3 1

=

> .

37. [R] Find a basis for R4 which contains a basis of im(D), where

D =

;

A

A

<

1 3 3 #1 72 6 3 1 83 9 3 4 74 12 0 8 4

=

B

B

>

.

38. [R] A linear map T : R4 ' R4 has the property that

T

;

A

A

<

1000

=

B

B

>

=

;

A

A

<

3000

=

B

B

>

, T

;

A

A

<

0100

=

B

B

>

=

;

A

A

<

4000

=

B

B

>

, T

;

A

A

<

0010

=

B

B

>

=

;

A

A

<

#1300

=

B

B

>

, T

;

A

A

<

0001

=

B

B

>

=

;

A

A

<

0#301

=

B

B

>

.

39


a) Write down the matrix representation of T with respect to the standard basis (inboth domain and co-domain).

b) Find a basis for the image of T and find the rank of T .

c) State the dimension of the kernel of T .

d) Does the vector

;

A

A

<

8#312

=

B

B

>

belong to the image of T ? Give reasons.

39. [H] Let A " Mnn(R). Show that the following statements are equivalent, that is, show that ifany statement is true then all are true, whereas if any statement is false then all are false.

a) For all x and y in Rn, Ax = Ay if and only if x = y.

b) ker(A) = {0}.c) nullity(A) = 0.

d) rank(A) = n.

e) im(A) = Rn.

f) The columns of A form a basis for Rn.

40. [H] Let A " Mmn(R), and let {ej : 1 ! j ! m} be the set of m standard basis vectors of Rm.If A is of rank r, explain why at most r of the m equations Axj = ej can have solutions.

41. [H] Let A and ej be as in the previous question. If nullity(A) = (, explain why at leastm# n+ ( of the m equations Axj = ej do not have solutions.

42. [H] Let A " Mnn(R), rank(A) = n, and ej be the standard basis vectors for Rn. Prove thateach of the n equations Axj = ej , 1 ! j ! n, has a unique solution.

43. [H] Let T : V ' V be a linear map and assume that dim(V ) = n. Show that the followingstatements are equivalent.

a) T (v) = T (w) if and only if v = w for all v,w " V .

b) ker(T ) = {0}.c) nullity(T ) = 0.

d) rank(T ) = n.

e) im(T ) = V .

Problems 8.5

44. [R] Show that the function T : R4 ' M22(R) defined by

T (a) =

%

a1 a2a3 a4

&

for a = (a1, a2, a3, a4) " R4

is a linear map.

40


45. [R] Show that the function T : R4 ' M22(R) defined by

T (a1, a2, a3, a4) =

%

3a1 # 2a4 a4 + 2a3#5a2 + 3a3 a1

&

is a linear map.

46. [R] Is the function T : M23(R) ' R6 defined by

T (A) ="

a11 a12 a13 a21 a22 a23#T

for A =

%

a11 a12 a13a21 a22 a23

&

" M23(R)

a linear map?

47. [R] Show that the function T : P2 ' C3 defined by

T (a0 + a1z + a2z2) =

;

<

a0a1a2

=

>

is a linear map. Note that T maps a polynomial in P2 into its coordinate vector withrespect to the standard basis {1, z, z2}.

48. [R] Show that the function T : C4 ' P4 defined by T (a) = p for a =

;

A

A

<

a1a2a3a4

=

B

B

>

" C4, where

p(z) = (a1 # 3a2) + (2a3 # 3a4)z + a2z3 + (3a1 # a2 + 2a3 + 4a4)z

4 for all z " C,

is a linear map.

49. [H] Show that the function T : P3(R) ' P3(R) defined by

T (p) = 4p$ + 3p, where p$(x) =dp

dx,

is a linear map.

50. [H] Is the function T : P3(R) ' P3(R) defined by T (p) = q, where

q(x) = 4xp$(x)# 8p(x) for x " R,

a linear map? Prove your answer.

51. [H] Show that the function T : P3(R) ' P4(R) defined by T (p) = q, where

q(x) =

2 x

0p(t)dt for x " R,

is a linear map.

41


52. [H] Let V be the subset of the vector space R[R] of all real-valued functions on R defined by

V =

C

f " R[R] :

2 x

0f(t)dt exists for all x " R

D

.

Show that V is a subspace of R[R], and then show that the rule T : V ' R[R] defined byT (f) = g, where

g(x) =

2 x

0f(t)dt for f " V and x " R,

is a linear map.

53. [H] A function S : R ' Z is defined by S(x) = y, where y is the integer obtained on roundingx to the nearest integer. Is S a linear map? A function T : R ' R is defined by T (x) = y,where y is the integer obtained on rounding x to the nearest integer. Is T a linear map?

54. [H] Let y be a real-valued function with domain R such that y and its first two derivatives y$

and y$$ exist, and such that the Laplace transforms (see Example ?? of Section 8.5) of y,y$ and y$$ also exist on the interval (0,&). Given that y(0) = 1 and y$(0) = 2 and that ysatisfies the di"erential equation

y$$(x) + 4y$(x) + 3y = e!3x,

find an explicit formula for the Laplace transform yL(s) of y in terms of s.

HINT. Take the Laplace transform of the di"erential equation and use integration by partsto find formulae for the Laplace transforms of y$ and y$$ in terms of yL(s).

55. [H] Consider the mapping T : P3(R) ' R2 defined by

T (p(x)) =

%

a# bc# d

&

where p(x) = a+ bx+ cx2 + dx3.

a) Prove that T is linear.

b) Show that p(x) = 3x3 + 3x2 # 2x# 2 is in the kernel of T .

56. [H] Consider the function T : R4 ' P1 defined by

T

;

A

A

<

abcd

=

B

B

>

= (a# 2b) + (c+ d)x.

a) Find T

;

A

A

<

1#32

#4

=

B

B

>

.

b) Show T is a linear transformation.

42


c) Write down a non-zero vector in R4 which lies in ker(T ).

57. [H] A linear map T : C3 ' P3 has function values given by

T

;

<

100

=

> = 1 + (2 + i)z # 3z3, T

;

<

010

=

> = (4# 3i)z + z2, T

;

<

001

=

> = #2 for z " C.

Find T

;

<

i2#1

=

> and T

;

<

x1x2x3

=

>.

58. [H] Let Pn be the real vector space of polynomials of degree less than or equal to n, and takeits standard basis to be

{1, x, x2, . . . , xn}.

For p(x) " P3, let (T (p)) (x) " P4 be defined by

(T (p)) (x) =

2 x

0p(t)dt.

a) Show that T is a linear transformation from P3 to P4.

b) Calculate the matrix A of this linear transformation with respect to the standardbases of P3 and P4.

c) Find a basis for the image of T , im(T ).

d) Find a basis for the kernel of T , ker(T ).

59. [R] A car manufacturer produces a station wagon, a four-wheel drive, a hatchback and a sedanmodel. Each model is made from steel, plastics, rubber and glass, and it also requires anumber of hours of labour to produce. The requirements per car of these inputs for eachmodel are as shown in the following table.

Steel Plastics Rubber Glass Labour(tonnes) (tonnes) (tonnes) (tonnes) (hours)

Station wagon 1 0.5 0.1 0.2 14-wheel drive 1.5 0.6 0.2 0.15 1.5hatchback 0.8 0.7 0.2 0.2 1.1sedan 0.9 0.6 0.25 0.3 0.9

Construct a matrix which can be used to express the factory input as a linear function ofthe factory output.

Problems 8.6

60. [R] Let idR2 be the identity map for R2. Find a matrix representation of this map with respectto standard bases in domain and codomain.

43


61. [H] Let idR2 be the identity map for R2. Find a matrix representation of this map with respect

to the domain basis

C%

11

&

,

%

1#1

&D

and the codomain basis

C%

13

&

,

%

12

&D

.

62. [H] A linear mapping G : P2 ' P2 has matrix representation

A =

;

<

1 2 #1#1 1 03 4 1

=

>

with respect to the standard basis {1, x, x2} in both domain and co-domain. Find G(p),where p(x) = #3 + x+ 5x2.

63. [H] For each of the linear maps in Questions 48 to 51, find a matrix which represents the linearmap for standard bases in the domain and codomain.

64. [H] Using your results of the previous question or otherwise, find the kernel, nullity, image andrank of the linear maps in Questions 48 to 51.

65. [H] For the linear map T : Pn(R) ' Pn(R) defined by T (p) = q, where

q(x) = x2d2p

dx2# 3x

dp

dx+ 3p(x) for x " R,

find a matrix which represents T with respect to standard bases in domain and codomain.Hence, or otherwise, find the kernel and nullity of T .

66. [H] Let V be a vector space and let B = {u1,u2,u3} be an orthonormal basis for V . Let

a " V be a vector whose coordinate vector with respect to B is

;

<

a1a2a3

=

>. Let

;

<

a$1a$2a$3

=

> be the

coordinate vector of a with respect to the basis B$ = {u$1,u

$2,u

$3} given by

u$1 =

1$2u1 +

1$2u3,

u$2 = # 1$

2u1 +

1$2u3,

u$3 = #u2

Show that B$ is an orthonormal basis, then show that the rule T

;

<

a1a2a3

=

> =

;

<

a$1a$2a$3

=

> is a

linear map from R3 to R3, and find a matrix representation for this function with respectto standard bases for R3 in domain and codomain. Finally, show that the matrix you haveconstructed is a matrix representation of the identity map idV : V ' V with respect tothe basis B in domain and B$ in codomain.

44


67. [H] Suppose T is the linear transformation with matrix with respect to standard bases givenby

;

<

1 2 3 42 #4 6 #2

#1 2 #3 #1

=

> .

Find the matrix of T with respect to the bases

B1 =

,

-

-

.

-

-

/

;

A

A

<

1000

=

B

B

>

,

;

A

A

<

0100

=

B

B

>

,

;

A

A

<

#6#504

=

B

B

>

,

;

A

A

<

#3010

=

B

B

>

6

-

-

7

-

-

8

in the domain, and

B2 =

,

.

/

;

<

12#1

=

> ,

;

<

2#42

=

> ,

;

<

010

=

>

6

7

8

in the codomain.Can you explain where these bases have come from?

68. [H] Suppose T is the linear transformation with matrix with respect to standard bases givenby

;

<

3 0 00 #4 #10 6 3

=

> .

Find the matrix of T with respect to the basis

B =

,

.

/

;

<

0#16

=

> ,

;

<

0#11

=

> ,

;

<

100

=

>

6

7

8

in both domain and codomain.Can you explain what is going on here?

Problems 8.7

69. [H] Let T : V ' W and S : V ' W be linear maps. Let BV be a basis for V and BW be abasis for W and let A and B be the matrices representing T and S with respect to thebases BV and BW . Prove that A +B is the matrix representing the sum function T + Swith respect to the bases BV and BW .

70. [H] Let T : U ' V and S : V ' W be linear maps. Let BU , BV and BW be bases for U , Vand W respectively. Let A be the matrix representing T with respect to bases BU and BV

and let B be the matrix representing S with respect to bases BV and BW . Prove that thematrix product BA is the matrix which represents the composition function S,T : U ' Wwith respect to the bases BU and BW .

45


Problems 8.8

71. [H] Let V = C[R], the vector space of all continuous real-valued functions on R. Let

B = {ex, (x# 1)ex, (x# 1)(x # 2)ex} - V

a) Prove that B is linearly independent.

b) Let W = span (B), and let D : W ' W denote the linear transformation

D(f) = f $,

where f $ is the derivative of f.

i) Find the matrix for D with respect to the ordered basis B of W.

ii) Find the matrix for the linear transformation T = D ,D.

iii) Hence or otherwise, prove that for every g " W, there exists f " W such that

f $$ = g.

72. [H] For a field F, define T : M22(F) ' F3 by

T

%E

a11 a12a21 a22

F&

=

;

<

a22a12 # a213a11 + a12

=

> .

a) Show that T is a linear transformation.

b) Find the kernel of T and the nullity of T .

c) Find the rank of T .

d) Is T one-to-one (injective)? Give a brief reason for your answer.

e) Find the matrix of T with respect to the standard bases of M22(F) and F3.

46

Chapter 9

EIGENVALUES ANDEIGENVECTORS

9.1 Definitions and examples

9.2 Eigenvectors, bases, and diagonalisation

9.3 Eigenvalues of matrices with special structure

9.4 Applications of eigenvalues and eigenvectors

9.5 Markov systems

9.6 Eigenvalues and Matlab

47

48 CHAPTER 9. EIGENVALUES AND EIGENVECTORS


Problems 9.1

1. [R] Let

A =

%

3 00 #4

&

, B =

%

2 00 2

&

, C =

%

#3 00 0

&

,

and let e1 and e2 be the standard basis vectors for R2.

a) Write down the eigenvalues and eigenvectors of A, B and C.

b) Draw a sketch of e1, Ae1, e2, Ae2. Then, for some vector x which is not parallel toeither e1 or e2 draw a sketch of x and Ax.

c) Repeat part (b) for the matrix B. Comment on any di"erences you observe betweenthe results for A and B.

d) Repeat part (b) for the matrix C. Again comment on any di"erences you observebetween the results for A and C.

e) For x %= 0, prove algebraically that Ax is parallel to x if and only if x is parallel toeither e1 or e2, that Bx is parallel to x for all x and that Cx is parallel to e1 for allx.

2. [R] Show that the vector

%

11

&

is an eigenvector of the matrix

%

5 #32 0

&

and find the corre-

sponding eigenvalue.

3. [H] Let A be a fixed 3! 3 matrix and define a linear map T : M33 ' M33 by T (X) = AX. If& is a real eigenvalue of T corresponding to an invertible eigenvector X, find & in terms ofdet(A).

4. [H] Let T be the linear map which reflects vectors in R2 about the line y = x.

a) Explain why

%

11

&

and

%

1#1

&

are eigenvectors of T and give their corresponding

eigenvalues.

b) Find the matrix A such that Tx = Ax for all x " R2.

5. [R] Find the eigenvalues and eigenvectors for

a) A =

%

6 #26 #1

&

, b) A =

%

#5 2#6 3

&

.

6. [H] For each of the matrices in the preceding question find two independent eigenvectors v1

and v2. On one diagram sketch the lines

'1 = {x : x = µv1, µ " R}'2 = {x : x = µv2, µ " R}

48


and the parallelogram

P = {x : x = µ1v1 + µ2v2, for 0 ! µ1 ! 1, 0 ! µ2 ! 1}.

Then identify and sketch (on a separate diagram)

{y : y = Ax, x " '1}{y : y = Ax, x " '2}{y : y = Ax, x " P}.

Describe the linear mapping Tx = Ax geometrically.

7. [R] Find the eigenvalues and eigenvectors of the following matrices. In each case, note if theeigenvalues are real, occur in complex conjugate pairs, or are general complex numbers.Also note if the eigenvectors form a basis for C2.

a)

%

1 22 1

&

, b)

%

2 10 2

&

, c)

%

3 50 #6

&

,

d)

%

0 #21 2

&

, e)

%

4 2i2i 6

&

, f)

%

4 #2i2i 6

&

.

8. [H] Show that the eigenvalues of a square row-echelon form matrix U are equal to the diag-onal elements of the matrix. (A square row-echelon form matrix is also called an uppertriangular matrix).

9. [R] Find the eigenvalues and eigenvectors of the row-echelon matrix

U =

;

A

A

<

2 #4 1 30 #2 1 #30 0 3 30 0 0 5

=

B

B

>

.

10. [R] Find the eigenvalues and eigenvectors of the following matrices.

a) A =

;

<

1 3 02 2 00 0 6

=

>. b) B =

;

<

3 0 00 #4 #10 6 3

=

>.

Problems 9.2

11. [R] For each of the matrices in Questions 7, 9 and 10, decide if the matrix is diagonalisable,and if it is find an invertible matrix M and a diagonal matrix D such that D = M!1AM .

12. [H] Show that if & is an eigenvalue of A then & is also an eigenvalue of the matrix A$ = B!1AB,where B is any invertible matrix. Also show that if v is an eigenvector of A for eigenvalue& then B!1v is an eigenvector of A$ for eigenvalue &.

49


13. [H] Show that if & is an eigenvalue of A then & is also an eigenvalue of AT .

HINT: Use the characteristic equation and the properties of determinants.

14. [H] Let A be an n! n matrix. Let TA : Cn ' Cn be the linear transformation defined by

TA(x) = Ax for x " Cn.

Let the columns of an n! n matrix B be an ordered basis for Cn. Show that the matrixrepresenting TA with respect to the basis formed by the columns of B is B!1AB.

HINT. The method used in Example ?? of Section 8.6 might be helpful.

NOTE. Modern methods of finding eigenvalues search for a change of basis which makesA$ = B!1AB into an upper triangular matrix. As shown in Question 8 the eigenvaluesare then the diagonal elements of the upper triangular matrix. The actual algorithms forfinding the change of basis are complicated.

15. [H] Let T : V ' V be linear. Show that if B is any basis for V and A is the matrix representingT with respect to the basis B in both domain and codomain of T then the eigenvalues ofT and A are the same. What is the relation between the eigenvectors of T and A?

Problems 9.3

16. [R] Find an orthogonal matrix Q which diagonalises

%

1 22 1

&

.

17. [R] Show that

;

<

#201

=

> ,

;

<

110

=

> and

;

<

#11#2

=

> are eigenvectors of

A =

;

<

0 #1 2#1 0 #22 #2 3

=

>

and hence find an orthogonal matrix Q which diagonalises A.

Problems 9.4

18. [R] Let A =

%

0 61 #1

&

. Diagonalise A and hence find A5.

19. [R] Let A =

%

0 38 2

&

.

a) Find the eigenvalues and eigenvectors of A.

50


b) Find matrices P and D such that

A = PDP!1.

c) Write down an expression for An in terms of P and D. Hence evaluate AnP.

20. [R] A first-order linear di"erence equation (often called a first-order linear recurrence relation)is an equation of the form

x(k + 1) = Ax(k), where k = 0, 1, 2, . . . ,

and where A is a fixed matrix.

The solution of this equation isx(k) = Akx(0),

as you can check by direct substitution. For the diagonalisable matrices of Questions 7, 9and 10, find Ak and hence evaluate x(k).

21. [R] For each of the diagonalisable matrices of Questions 5, 9 and 10 find general solutions ofthe di"erential equations

dy

dt= Ay.

22. [H] Solvedy

dt=

%

2 1#1 2

&

y.

23. [R] a) Find the eigenvalues and eigenvectors of

%

2 31 4

&

.

b) Hence solve the system of di"erential equations:

,

-

-

.

-

-

/

dx1dt

= 2x1 + 3x2,

dx2dt

= x1 + 4x2.

24. [R] Solve the following systems of di"erential equations, given that x(0) = y(0) = 100.

a)

,

-

-

-

.

-

-

-

/

dx

dt= 5x# 8y

dy

dt= x# y

b)

,

-

-

-

.

-

-

-

/

dx

dt= 3x# 15y

dy

dt= x# 5y

25. [R] Solve the following second-order linear di"erential equations with constant coe!cients bythe “calculus method” and by the matrix method and compare your answers.

a) 5d2y

dt2# 6

dy

dt+ y = 0.

51


b)d2y

dt2# 16y = 0.

26. [R] What happens if you try to solve the second-order equation

d2y

dt2+ 4

dy

dt+ 4y = 0

by the matrix method?

27. [H] Consider the second-order linear di"erential equation

ad2y

dt2+ b

dy

dt+ cy = 0,

where a, b, c " R and a %= 0.

a) Assume that the solutions to the characteristic equation a&2 + b& + c = 0 for thissecond-order di"erential equation are distinct. By making the substitutions y1 = y

and y2 =dy1dt

, convert the di"erential equation into a system of first-order linear

di"erential equations

dy

dt= Ay, where y =

%

y1y2

&

.

b) Using matrix methods, show that the general solution of this system is

y = "1e%1t

%

1&1

&

+ "2e%2t

%

1&2

&

.

Compare this solution with that obtained using the usual “calculus method” of solvingthe original second-order linear di"erential equation.

28. [H] A radioactive isotope A decays at the rate of 2% per century into a second radioactiveisotope B, which in turn decays at a rate of 1% per century into a stable isotope C.

a) Find a system of linear di"erential equations to describe the decay process. If westart with pure A, what are the proportions of A, B, and C after 500 years, after1000 years, and after 1000000 years?

First solve this problem using matrix methods, and then try to solve the problemdirectly by solving the original two di"erential equations in the right order.

b) Explain how the problem would be di"erent if the rates of decay of A and B wereboth 2% per century.

29. [H] Algebraically consider the behaviour of x(k) as k ' &, where

x(k + 1) =

%

0.9 0.20.1 0.8

&

x(k)

and x(k) is a probability vector.

52


30. [H] There are 3 mathematics lecturers A, B and C who are teaching parallel streams in al-gebra to a total of 900 students. At the first lecture equal numbers go to each lecturegroup. After each lecture a certain percentage of the students in each group decide to staywith the same lecturer while the remaining percentage divide evenly among the other twolecturers for the next lecture. If 98% of A’s students stay with A each time, 96% of B’sstudents stay with B and 94% of C’s students stay with C, find the numbers of studentsin each group in the 12th lecture and in the 24th lecture. Make the assumption that nostudents stop attending lectures.

HINT. Set up a model as a di"erence equation of the type given in Question 20. You mayuse Matlab to find all eigenvalues and eigenvectors. Alternatively, if you wish to solvethe problem by hand calculations, you will need to know that one of the eigenvalues is 1.

NOTE. This problem is an example of a Markov chain process. Markov chain processes areimportant in many areas of mathematics and its applications, such as statistics, psychol-ogy, finance, economics, operations research, queueing theory, inventory theory, di"usionprocesses, theory of epidemics etc.

31. [H] Repeat the previous question on the following assumptions. After each lecture, 1% of eachgroup stop attending lectures altogether, and the remaining percentage either stay withthe same lecturer or divide equally among the other two lecturers for the next lecture. If97% of A’s students stay with A each time, 95% of B’s students stay with B and 93% ofC’s students stay with C, find the numbers of students in each group in the 12 lecture andin the 24th lecture. Also find the total number of students attending lectures in the 12thlecture and in the 24th lecture.

32. [H] Consider a modified version of the population dynamics model of Example ?? of Sec-tion 8.5, in which all females are assumed to die at age 74 instead of at age 89, as in themodel given. Use eigenvalues and eigenvectors to solve this modified model, given thatthere are one million females in each age group at January 1, 1970. What happens to thepopulation for large values of k?

NOTE. You will need to use Matlab to find the eigenvalues and eigenvectors of the matrixA, and you may also use Matlab, if you wish, to carry out all other matrix manipulationsrequired to solve the problem.

Problems 9.5

33. [H] Let A be a 2 ! 2 matrix with the property that all its entries are non-negative and bothits columns sum to 1. Show that &1 = 1 is always an eigenvalue for A, and that if &2 isanother eigenvalue of A then #1 . &2 ! 1.

34. [H] Show that the matrix of the original population dynamics model of Example ?? of Sec-tion 8.5 is not diagonalisable.

HINT. Use Matlab to find the eigenvalues, and then show that the eigenvalue & = 0 hasmultiplicity 2 and that dim(ker(A)) = 1, i.e., a basis for the kernel of A # 0I consists of

53


one vector.

NOTE. The original population dynamics model can be solved by a generalisation of theeigenvalue-eigenvector methods which makes use of “Jordan forms”, and is covered in oursecond year linear algebra subjects.

54

ANSWERS TO SELECTEDPROBLEMS

Chapter 6

1.x " N x " Z x " Q x " R

a) - #25 #25 #25

3 3 3 3

- #3 #3 #3

- - #103 #10

3

b) 1 1,#5 1,#5 1,#5

5 5 5,32 5,32- - - 1±

%5

2

- - - -

c) 3j,j " N 3j,j " Z 3j,j " Z 3j,j " Z

0 0 0 3k!,k " Z

2. No.

3. Yes. The set {0} and the empty set / = { }.

4. Yes.

5. 3z = 6 + 9i, z2 = #5 + 12i, z + 2w = 7i, z(w + 3) = #2 + 10i,z

w=

1

5(4 # 7i),

w

z=

1

13(4 + 7i).

6. a)1

5(3# i), b) #1

2(1# i).

7. a) a2 # b2 +2abi, b)a

a2 + b2# i

b

a2 + b2, c)

1

(a# 1)2 + b2"

(a2 # 1 + b2)# 2ib#

.

55

56 CHAPTER 6

8. a) 12 (#1±

$3 i), b) #1±

$2 i, c) 3± i, d) 1

2 (3±$3), e) ±i, ±2i.

10. 16

11.8abi(a2 # b2)

(a2 + b2)2

12.

z Re(z) Im(z) z

#1 + i #1 1 #1# i

2 + 3i 2 3 2# 3i

2# 3i 2 #3 2 + 3i

2!i1+i

12 #3

21+3i2

1(1+i)2 0 #1

212

13. #3 + 4i,11

25# 2

25i.

14. z = 2 + 3i, w = #1 + 2i.

17. b) z2 # 6z + 13

18.

z |a| Arg(z) Polar Form

6 + 6i 6$2 !

4 6$2"

cos !4 + i sin !

4

#

#4 4 ! 4 cos ! + i sin !)$3# i 2 !!

6 2"

cos !6 # i sin !

6

#

!1%2# i%

21 !3!

4 cos 3!4 # i sin 3!

4

#7 + 3i$58 "

$58(cos"+ i sin") Here " = ! # tan!1 3

7 .

19.$234, #1.

20. n = 4

21. a)3

2

"

1 +$3i

#

, b)3

2

"

#$3 + i

#

, c) #3

2

"

1 +$3i

#

, d)3

2

"$3# i

#

,

e)3

2

'

G

2 +$2 + i

G

2#$2(

(Double angle formula used).

27. 64, #(1 +$3)i,

1 +$3

2+

$3# 1

2i.

28.7

2.

56

ANSWERS 57

29. !.

30. Arg (#1 + i) =3!

4; Arg

"

#$3 + i

#

=5!

6;

Arg"

(#1 + i)"

#$3 + i

##

= #5!

12; Arg

%

#1 + i

#$3 + i

&

= # !12

.

31. sin7!

12=

1 +$3

2$2

.

32. zw = 2$2ei!/12 = 2

$2H

cos' !

12

(

+ i sin' !

12

(I

; z9 = #512;' z

w

(12= 64ei! = #64.

33. a) 16"

#$3 + i

#

, b) # i, c) # 1

2# i

$3

2.

34. a) ±(5# 2i), b) ±(3 + 5i), c) ±(7 + 5i).

35. b)$2e

5!i

12 = 12

""$3# 1

#

+ i"$

3 + 1##

, c) 1%2(1 + 13i).

37. a) 2 + i, 1# i; b) 4 + i, 3# 2i; c) 1# 2i, #5 + 3i.

38. ei!/7, e3i!/7, e5i!/7, ei!, e!i!/7, e!3i!/7, e!5i!/7.

39. ein!/12 for n = #11,#7,#3, 1, 5, 9.

40. 2ein!/15 for n = #13,#7,#1, 5, 11.

41.15

2+ i

)

3$3

2# 1

*

, 3# i,15

2# i

)

3$3

2+ 1

*

.

48. a) Real part = cos(2#). Imaginary part = sin(2#).

51. a) cos 6# = cos6 # # 15 cos4 # sin2 # + 15 cos2 # sin4 # # sin6 #sin 6# = 6cos5 # sin # # 20 cos3 # sin3 # + 6cos # sin5 #

b) cos 6# = 32 cos6 # # 48 cos4 # + 18 cos2 # # 1

52. sin 7# = 7cos6 # sin # # 35 cos4 # sin3 # + 21 cos2 # sin5 # # sin7 #cos 7# = cos7 # # 21 cos5 # sin2 # + 35 cos3 # sin4 # # 7 cos # sin6 #.

53. a) cos # =1

2

"

ei" + e!i"#

.

b) cos6 # =1

32(cos 6# + 6cos 4# + 15 cos 2# + 10).

54. sin5 # =1

16(sin 5# # 5 sin 3# + 10 sin #)

2

sin5 #d# =1

16

%

#1

5cos 5# +

5

3cos 3# # 10 cos #

&

+ C,

57

58 CHAPTER 6

cos4 # =1

8[3 + 4 cos(2#) + cos(4#)]

2

cos4 #d# =1

8

E

3# + 2 sin(2#) +1

4sin(4#)

F

+ C.

55. a) cos 5# = 16x5 # 20x3 + 5x

d) #1, cos !5 , cos

3!5 , cos 7!

5 , cos 9!5 .

56. The sum is n when k is an integer multiple of n and 0 otherwise.

58.sin

"

12(n+ 1)#

#

sin"

12n#

#

sin 12#

.

59. a)9ei"

9 + ei2".

60. a) |z # i| ! 2 b) |z # i| ! 2 or # !3! Arg (z # i) !

2!

3

i2

i

c) |z| " 2 and |Im(z)| ! 3 d) y ! x

y = 3

y = #3

20

y=x

0

58

ANSWERS 59

e) The real axis f) |z # 1# i| < 1 & #!4< Arg (z # 1# i) !

!

2

0

Im

Re0

1 + i

g) Circle: x2 +

%

y +5

3

&2

=

%

4

3

&2

h) Ellipse:

%

x

2$2

&2

+'y

3

(2= 1

#3i

#53 i

0

#3i

3i

#2$2 2

$20

61. a) Re(z) " 3 Im(z) and |z # (3 + i)| > 2

1

3 S1

Real Axis

x = 3y

Im Axis

Circle radius 2, centre 3 + i

0

59

60 CHAPTER 6

b) |z # i| < |z + i| and #!6! Arg (z # i) !

!

6

1

Im Axis

Real Axis

|z # i| < |z + i| is equivalent toIm(z) > 0

!6!6

S2

0

62. a) Im(z) > #4 and |z # 1# i| " 3 b) Yes.

1 + i

3

Re(z)

Im(z)

Im(z) = #4

63. | z # x |"| z # Re(z) |

Re(z) x

z

| z #x|

|z#

Re(z)

|

0

60

ANSWERS 61

64. a) w = ei#, #! < " ! ! c) | z # ei# |"| z # ei" |, # = Arg(z)

Real axis#

1

Real axis"

#

z|z !

ei" |

|z!ei#|

65. a) 742, b) 129, c) 1 + 9i.

66. p(z) = (z # 2)(2z # 5)(z + 3).

67. p(z) = (z # 1)(z + 1)(z + 2)(z + 4).

68. a)'

z # e!i!

10

( '

z # e3!i

10

('

z # e7!i

10

('

z # e!i!

2

( '

z # e!9!i

10

(

.

b)'

z #$2e

i!

6

('

z #$2e

i!

2

('

z #$2e

5!i

6

( '

z #$2e!

i!

6

( '

z #$2e!

i!

2

('

z #$2e!

5!i

6

(

.

69. a) (x# 1)(x+ 1)(x2 + 1)(x2 +$2x+ 1)(x2 #

$2x+ 1).

b) (x2 + 2)(x2 +$6x+ 2)(x2 #

$6x+ 2).

70. (z2 + 2z + 2)(z2 # 2z + 2)

71. (z # e!i!/8) (z # ei3!/8) (z # ei7!/8) (z # ei11!/8)

72. a) e!5!i

6 , e!!i

2 , e!!i

6 , e!i

6 , e!i

2 , e5!i

6 .

b) Note that the solutions are evenly spaced around the unit circle centred on 0.

Re z

Im z

0

i

e5i!

6

e!5i!

6

#i

e!i!

6

ei!

6

c)

%

z # e!5!i6

&%

z # e!!i2

&%

z # e!!i6

&%

z # e!i6

&%

z # e!i2

&%

z # e5!i6

&

.

d)"

z2 + 1# "

z2 +$3 z + 1

# "

z2 #$3 z + 1

#

.

61

62 CHAPTER 7

73. a) ei!/4, ei!/2, ei3!/4, e!i!/4, e!i!/2, e!3!/4.

b)"

z # ei!/4# "

z # e!i!/4# "

z # ei3!/4# "

z # e!i3!/4# "

z # ei!/2# "

z # e!i!/2#

c)"

z2 #$2z + 1

# "

x2 +$2z + 1

# "

z2 + 1#

.

74. a) (z # e2i!/5)(z # e!2i!/5)(z # e4i!/5)(z # e!4i!/5)

b)

%

z2 # 2z cos

%

2!

5

&

+ 1

& %

z2 # 2z cos

%

4!

5

&

+ 1

&

75. a) (t+ 1# i) (t+ 1 + i) (t# 2) (t+ 1) (t+ i) (t# i),

b) (t2 + 2t+ 2) (t# 2) (t+ 1) (t2 + 1).

76. 1 + i, 1# i, 3$5,

3$5

2

"

#1 + i$3#

,3$5

2

"

#1# i$3#

.

77. a) (z2 + z + 1)(z6 + z3 + 1). b) e±2i!/9, e±4i!/9, e±8i!/9.

79. a) One of the roots is (#2 + 2i)1/3 + (#2# 2i)1/3.

80. d) #2, 2$2 cos

!

12, 2

$2 cos

7!

12.

81. a) 1; b) #1,5

7,1

4; c) 4, ±1

5.

83. a) Unstable. b) Neither decays nor grows. c) Unstable.

84. a) Stable. b) Unstable. c) Neither decays nor grows.

93. a) 2123i. b) i. c) No.

Chapter 7

3. a)

;

<

120

=

> ,

;

<

114

=

> .

7. Axioms 1, 4, 5, 6, 10 are satisfied, others are not. It is not a vector space.

8. a) 2v = (1 + 1)v = 1v + 1v = v+ v. Identify the axioms that have been used.

b) Use induction.

9. For part 5: If &v = µv then &v # µv = 0, so (& # µ)v = 0. But v %= 0 so, by part 4 of theproposition, &# µ = 0. So & = µ.

62

ANSWERS 63

10. a)

;

<

3#41

=

>. b) x =

;

<

1#12

=

> + &

;

<

2#3#1

=

> for & " R.

11. a)

;

<

1#12

=

> ,

;

<

2#20

=

>. b) x =

;

<

111

=

> + &

;

<

1#12

=

> for & " R.

c) x =

;

<

111

=

> + &

;

<

1#12

=

> + µ

;

<

2#20

=

> for &, µ " R. d)

;

<

440

=

>.

e)

;

<

440

=

> ·

;

<x#

;

<

111

=

>

=

> = 0. f) x+ y # 2 = 0.

12.$3,

$6, 4,

2$2

6.

13. a) 2x+ 4y # z = 11. b)

;

<

24#1

=

> ·

;

<

;

<

xyz

=

> #

;

<

00

#11

=

>

=

> = 0

14.

;

<

011

=

> ,

$2

2.

15. a)

;

<

2#1#2

=

>, b) #1

9

;

<

2#1#2

=

>, c)1

3.

17. The position vectors of the points on a plane which does not pass through the origin do notform a vector subspace.

18. a) For example,

;

<

000

=

> ,

;

<

201

=

> ,

;

<

12#2

=

> are in S

c) The position vectors of the points on a plane which passes through the origin form avector subspace.

22. Column 1 belongs to S as A

;

<

100

=

> =

%

24

&

. Similar arguments apply for columns 2 and 3.

23. a) No. b) Yes. c) Yes.

63

64 CHAPTER 7

26. W =

,

-

-

-

-

.

-

-

-

-

/

;

A

A

A

A

<

00")*

=

B

B

B

B

>

: ",), * " R

6

-

-

-

-

7

-

-

-

-

8

, a copy of R3.

29. No, S is not a subspace because the zero polynomial is not in it.

30. b) x3 + 3x2 + 3x+ 1.

31. HINT: Suppose that S1 =

C%

x0

&

: x " R

D

and S2 =

C%

0y

&

: y " R

D

. Show that S1 and S2

are subspaces of R2 but S1 0 S2 is not.

34. a) Yes, a = 2v1 # 3v2 + v3. b) Yes.

35. a) No. b) No, 3b2 # 2b3 = 0. span(v1,v2,v3) is a plane in R3.

36. a) Yes, a = v1 # v2 # 2v3

b) No, b " span(v1,v2,v3) if and only if b1 # 2b2 + b4 = 0.The span is a 3-dimensional subspace in R4.

37. Yes, b = #10v1 + 12v2 + 16v3.

38. No, span(v1,v2,v3) is plane in R3 given by 5b1 # 4b2 + b3 = 0.

39. Yes. v = 3v1 # v2 # 2v3, where v1,v2,v3 are the columns of A.

40. No.

41. Yes.

44. Linearly dependent, coplanar.

45. Linearly independent, not coplanar.

46. Set containing 0 is linearly dependent as coe!cient of zero vector can be varied to make linearcombination non-unique.

47. b) v3 = v1 + v2.

e) span(S) is plane through origin parallel to v1 and v2.

48. b) v4 = #2v1 + 2v2 + v3 e) span (S) = R3.

49. No. #2# x+ 5x2 = (1# x+ 2x2) + 3(#1 + x2).

50. No, it is impossible to return to origin.

64

ANSWERS 65

51. Yes, it is possible to return to origin.

53. Yes, it is a basis.

54. A basis for W is

,

.

/

;

<

123

=

> ,

;

<

11#1

=

>

6

7

8

. Dim(W ) = 2.

56. a) True. b) False. c) False. d) True. e) False, True, False, False.

f) False. g) True. h) False. i) True. j) True.

57. a) n ! ' b) No relation. c) n " '. d) n = '.

59. b)

,

.

/

;

<

#110

=

> ,

;

<

#101

=

>

6

7

8

.

60. A basis for col(A) is

,

-

-

.

-

-

/

;

A

A

<

1000

=

B

B

>

,

;

A

A

<

#1100

=

B

B

>

,

;

A

A

<

#2420

=

B

B

>

6

-

-

7

-

-

8

. Dim(col(A)) = 3.

61. A basis for col(A) is

,

-

-

.

-

-

/

;

A

A

<

10#11

=

B

B

>

,

;

A

A

<

0#110

=

B

B

>

,

;

A

A

<

2#244

=

B

B

>

6

-

-

7

-

-

8

. Dim(col(A)) = 3.

62. A basis is

,

-

-

.

-

-

/

;

A

A

<

1234

=

B

B

>

,

;

A

A

<

3555

=

B

B

>

,

;

A

A

<

#7#8#32

=

B

B

>

,

;

A

A

<

0100

=

B

B

>

6

-

-

7

-

-

8

.

63. a) B = {v1,v3}. b) x = #4v1 + 5v3, c) dim(col(A)) = 2.

65. Bases are {p1, p2, p4} or {p1, p3, p4} or {p2, p3, p4}. {p1, p2, p3} is not a basis; why not?

68. Coordinate vector is

;

A

A

<

#24124

=

B

B

>

.

69. v =

;

A

A

<

1871119

=

B

B

>

.

65

66 CHAPTER 7

70. v =

;

<

1110

=

>.

71. a)

;

<

322

=

>. b)

;

<

#a1 # a2 + 2a3a2

#a1 + a3

=

>.

72. a)

;

<

12#821

=

>; b)

;

<

#21#3

=

>.

73. c) Coordinates are &1 =

;

<

#134

=

> · v1 = #2$2, &2 =

;

<

#134

=

> · v2 = 2$3,

&3 =

;

<

#134

=

> · v3 =$6. Coordinate vector is

;

<

#2$2

2$3$6

=

>.

75. a) For example,

%

1 00 4

&

,

%

1 23 4

&

,

%

#1 00 6

&

are in S.

b) S is not a subspace because the 2! 2 zero matrix is not in it.

76. a) For example,

%

1 00 #1

&

,

%

1 23 #1

&

,

%

0 00 0

&

are in S.

b) Use the Subspace Theorem to prove that S is a subspace.

80. a)

;

A

A

<

a11a12a21a22

=

B

B

>

. b)

;

A

A

A

A

<

12(a11 + a22)12(a12 + a21)

12i(#a12 + a21)12(a11 # a22)

=

B

B

B

B

>

.

81. a)

%

#4 2#1 #3

&

= #4

%

1 0#2 0

&

+ 2

%

0 13 0

&

# 3

%

0 05 1

&

. b) No.

85. Not a subspace.

91. Not a subspace.

93. No.

94. Let p " P2 be p(z) = a0 + a1z + a2z2. Then the condition is #32a1 + a2 = 0. (An equivalent

condition is p$"

#13

#

= 0.)

95. No for Question 93. No for Question 94.

66

ANSWERS 67

96. No. p3 = #p1 + 3p2.

97. A basis is {p1, p2, p3, p4, 1, z}.

98.

;

<

2#10

=

>.

99.

;

<

9117

=

>.

100.

;

<

a0 # a1 + a2a0

a0 + a1 + a2

=

>.

Chapter 8

1. S is not a linear map as the domain [#1, 1] is not a vector space.

2. a) Linear. b) Linear. c) Not Linear. d) Linear. e) Not Linear.

3. a) Domain C, codomain R, linear. b) Domain C, codomain R, linear.

c) Domain C, codomain R+ = {x " R : x " 0}, not linear.d) Domain C# {0}, codomain (#!,!], not linear. e) Domain C, codomain C, linear.

6. No.

7. T

;

<

2#14

=

> =

;

A

A

<

214

#1528

=

B

B

>

and T

;

<

x1x2x3

=

> =

;

A

A

<

x1 # 3x2 + 4x32x1

3x1 + x2 # 5x34x1 + 4x2 + 6x3

=

B

B

>

.

9.

;

<

021

=

> =

;

<

123

=

> +

;

<

#21#4

=

> +

;

<

1#12

=

>, and so T

;

<

021

=

> =

%

#11

&

.

11. a)

;

A

A

<

3 #12 4

#3 #30 1

=

B

B

>

. b)

;

<

#2 0 5 06 #8 0 2

#2 4 #3 0

=

>. d)

;

<

1 0 #2 #43 #4 #3 1

#2 2 4 0

=

>.

12. Same as for 11.

67

68 CHAPTER 8

13. a) Ae1 = 2e1, Ae2 = 0.7e2, Ab = 4e1 + 2.1e2. (e1 is stretched to twice its length, e2 iscompressed to 0.7 of its length and b is stretched and rotated.)

d) Notice that Ab = 3b. This means b is stretched to three times its length.

e) Notice that Ab = #2b. This means the direction of b is reversed and it is stretched totwice its length.

15.1

2

%

1 #$3$

3 1

&

.

16. x$ = T (x) =

%

#x1x2

&

, A =

%

#1 00 1

&

.

17. x$ = T (x) =

;

<

x1x2#x3

=

>, A =

;

<

1 0 00 1 00 0 #1

=

>.

18. q = Ap, where diagonal entries of A are aii = #1+2d2i /|d|2 and o"-diagonal entries of A areaij = 2didj/|d|2.

19. T is linear. For T (x) = Ax, the matrix is

A =

;

<

0 #b3 b2b3 0 #b1

#b2 b1 0

=

> .

20. T (a) = Aa, where A =1

5

;

<

1 0 20 0 02 0 4

=

>.

21. S is not linear.

23.

;

<

a$1a$2a$3

=

> = R#

;

<

a1a2a3

=

> =

;

<

a1 cos"+ a3 sin"a2

#a1 sin"+ a3 cos"

=

>.

25. a) ker(A) = {0}, nullity(A) = 0, no basis.

b) Kernel:

,

.

/

&

;

<

#1#31

=

> : & " R

6

7

8

, nullity(B) = 1.

c) Kernel:

,

-

-

.

-

-

/

&

;

A

A

<

2#2#11

=

B

B

>

: & " R

6

-

-

7

-

-

8

, nullity(C) = 1.

68

ANSWERS 69

26. a)

,

-

-

.

-

-

/

;

A

A

<

1#310

=

B

B

>

6

-

-

7

-

-

8

, nullity(D) = 1. b) {0}, nullity(E) = 0.

27. For example A =

%

1 1 1 11 2 3 4

&

.

28. a) ker(A) = {0}, nullity(A) = 0.

b) ker(A) =

,

-

-

.

-

-

/

&

;

A

A

<

5421

=

B

B

>

: & " R

6

-

-

7

-

-

8

, nullity(A) = 1.

d) ker(A) =

,

-

-

.

-

-

/

&

;

A

A

<

6411

=

B

B

>

: & " R

6

-

-

7

-

-

8

, nullity(A) = 1.

29. For Questions 16, 17 and 18, ker(T ) = {0} and nullity(T ) = 0.

For Question 19, ker(T ) = {x " R3 : x = &b for & " R}. Nullity(T ) = 1. Kernel is set of allvectors parallel to b.

For Question 20, ker(T ) = {x " Rn : b · x = 0}. Nullity(T ) = n# 1 (Why?). Kernel is set ofall vectors orthogonal to b.

30. b) 1

31. a) b " im(A) as A

;

<

2#31

=

> =

;

<

11104

=

> .

b) b is not in im(A) as Ax =

;

<

9#2#4

=

> has no solution.

c) b " im(A), since, for example, x =

;

A

A

<

#1012160

=

B

B

>

is a solution of Ax = b.

32. a) No conditions, im(A) = R3. b) 3b2 # 2b3 = 0.

c) No conditions, im(A) = R3.

33. rank(A) = 3. Columns 1,2,3 of A form a basis for im(A).rank(B) = 2. Columns 1,2 of B form a basis for im(B).rank(C) = 3. Columns 1,2,3 of C form a basis for im(C).

69

70 CHAPTER 8

rank(D) = 3. Columns 1,2,4 of D form a basis for im(D).rank(E) = 3. Columns 1,2,3 of E form a basis for im(E).

35. rank(A) = 3. Columns 1,3,4 of A form a basis for im(A).rank(B) = 3. Columns 1,3,4 of B form a basis for im(B).

36. One possible answer is

,

.

/

;

<

12#1

=

> ,

;

<

2#42

=

> ,

;

<

010

=

>

6

7

8

.

37. One possible answer is

,

-

-

.

-

-

/

;

A

A

<

1234

=

B

B

>

,

;

A

A

<

3330

=

B

B

>

,

;

A

A

<

#1148

=

B

B

>

,

;

A

A

<

1000

=

B

B

>

6

-

-

7

-

-

8

.

38. a)

;

A

A

<

3 4 #1 00 0 3 #30 0 0 00 0 0 1

=

B

B

>

. b)

,

-

-

.

-

-

/

;

A

A

<

3000

=

B

B

>

,

;

A

A

<

#1300

=

B

B

>

,

;

A

A

<

0#301

=

B

B

>

6

-

-

7

-

-

8

, 3. c) 1. d) No.

46. T is linear.

50. T is a linear function.

53. S is not linear as Z is not a vector space.T is not linear as, for example, T (1.5) + T (1.5) = 2 + 2 = 4 while T (1.5 + 1.5) = T (3) = 3.

54. yL(s) =s2 + 9s+ 19

(s+ 3)2(s+ 1).

56. a) 7# 2x c)

;

A

A

<

211#1

=

B

B

>

.

57. T

;

<

i2#1

=

> = (2 + i) + (7# 4i)z + 2z2 # 3iz3,

T

;

<

x1x2x3

=

> = (x1 # 2x3) + [(2 + i)x1 + (4# 3i)x2]z + x2z2 # 3x1z3.

58. b)

;

A

A

A

A

<

0 0 0 01 0 0 00 1

2 0 00 0 1

3 00 0 0 1

4

=

B

B

B

B

>

. c) {x, x2, x3, x4}. d) The empty set.

70

ANSWERS 71

59. Let input vector be b ="

b1 b2 b3 b4 b5#T

, where b1, b2, b3, b4 and b5 are the amountsof steel, plastics, rubber, glass and labour used respectively. Let the output vector be x ="

x1 x2 x3 x4#T

, where x1, x2, x3, x4 are the numbers of station wagons, 4-wheel drives,hatchbacks and sedans made. Then the factory is represented by the linear mapTA : R4 ' R5, where TA(x) = b = Ax with

A =

;

A

A

A

A

<

1 1.5 0.8 0.90.5 0.6 0.7 0.60.1 0.2 0.2 0.250.2 0.15 0.3 0.31 1.5 1.1 0.9

=

B

B

B

B

>

.

60. The matrix is

%

1 00 1

&

.

61. The matrix is

%

#1 #32 4

&

.

62. #6 + 4x

63. For 48,

;

A

A

A

A

<

1 #3 0 00 0 2 #30 0 0 00 1 0 03 #1 2 4

=

B

B

B

B

>

. For 49,

;

A

A

<

3 4 0 00 3 8 00 0 3 120 0 0 3

=

B

B

>

.

For 50,

;

A

A

<

#8 0 0 00 #4 0 00 0 0 00 0 0 4

=

B

B

>

; For 51,

;

A

A

A

A

A

A

A

<

0 0 0 01 0 0 0

0 12 0 0

0 0 13 0

0 0 0 14

=

B

B

B

B

B

B

B

>

.

64. 48: im(T ) = {p " P4(R) : p(z) = &0 + &1z + &2z3 + &3z4 for &0,&1,&2,&3 " R},(note z2 is not in im(T )) rank = 4, ker(T ) = {0}, nullity(T ) = 0.

49: im(T ) = P3(R), rank(T ) = 4, ker(T ) = {0}, nullity(T ) = 0.

50: im(T ) = {p " P3(R) : p(x) = &0 + &1x+ &2x3 for &0,&1,&2 " R}, rank(T ) = 3,ker(T ) = {p " P3(R) : p(x) = &x2 for & " R}, nullity(T ) =1.

51: im(T ) = {p " P4(R) : p(x) = &1x+ &2x2 + &3x3 + &4x4 for &1,&2,&3,&4 " R},rank(T ) = 4, ker(T ) = {0}, nullity(T ) = 0.

65. The matrix A is diagonal with diagonal elements akk = (k # 1)(k # 2) # 3(k # 1) + 3 for1 ! k ! n + 1. The kernel is "1x + "2x3 and the nullity is 2. Note that the kernel is thesolution of the homogeneous di"erential equation.

71

72 CHAPTER 9

66. The matrix is

;

A

A

<

1%2

0 1%2

# 1%2

0 1%2

0 #1 0

=

B

B

>

.

67.

;

<

1 0 0 00 1 0 00 0 0 0

=

>.

68.

;

<

2 0 00 #3 00 0 3

=

>.

71. b) i)

;

<

1 1 #10 1 20 0 1

=

>. ii)

;

<

1 2 00 1 40 0 1

=

>.

72. b) &

%

1 #3#3 0

&

, & " F; 1. c) 3. d) No. e)

;

<

0 0 0 10 1 #1 03 1 0 0

=

>.

Chapter 9

1. a) In each case, the eigenvalues are the diagonal entries and the respective eigenvectors arete1 and te2 (t %= 0).

For interpretations of (b), (c) and (d), see part (e) of question.

2. Eigenvalue is 2.

3. & = (detA)1/3.

4. b)

%

0 11 0

&

.

5. a) & = 2 with eigenvectors

C

t

%

12

&

: t %= 0

D

and & = 3 with eigenvectors

C

t

%

23

&

: t %= 0

D

,

b) & = #3 with eigenvectors

C

t

%

11

&

: t %= 0

D

and & = 1 with eigenvectors

C

t

%

13

&

: t %= 0

D

.

7. a) & = 3 with eigenvectors

C

t

%

11

&

: t %= 0

D

and

& = #1 with eigenvectors

C

t

%

#11

&

: t %= 0

D

.

72

ANSWERS 73

b) Only one eigenvalue & = 2 with multiplicity 2 and eigenvectors

C

t

%

10

&

: t %= 0

D

.

c) & = 3 with eigenvectors

C

t

%

10

&

: t %= 0

D

and

& = #6 with eigenvectors

C

t

%

#59

&

: t %= 0

D

.

d) & = 1± i with eigenvectors

C

t

%

#1± i1

&

: t %= 0

D

.

e) & = 5± i$3 with eigenvectors

?

t

)

±%32 + 1

2 i1

*

: t %= 0

@

.

f) & = 5±$5 with eigenvectors

C

t

%

12 (11

$5)i

1

&

: t %= 0

D

.

9. The eigenvalues are the diagonal entries, 2, #2, 3, 5. Corresponding eigenvectors are

v1 =

;

A

A

<

1000

=

B

B

>

, v2 =

;

A

A

<

1100

=

B

B

>

, v3 =

;

A

A

<

1150

=

B

B

>

, v4 =

;

A

A

<

25#32114

=

B

B

>

.

10. a) #1, 4, 6;

;

<

#320

=

> ,

;

<

110

=

> ,

;

<

001

=

> . b) 2,#3, 3;

;

<

0#16

=

> ,

;

<

0#11

=

> ,

;

<

100

=

> .

11. In each of the following answers, the diagonal entries in D and the columns in M may berearranged in the same way and the answer is still correct. Also, any column in M may bemultiplied by a scalar and the new M is still correct.

For Question 7:

a) D =

%

3 00 #1

&

, M =

%

1 #11 1

&

.

b) The matrix is not diagonalisable.

c) D =

%

3 00 #6

&

, M =

%

1 #50 9

&

.

d) D =

%

1 + i 00 1# i

&

, M =

%

1 + i 1# i1 1

&

.

e) D =

%

5 + i$3 0

0 5# i$3

&

, M =

)%32 + 1

2 i #%32 + 1

2 i

1 1

*

.

f) D =

%

5 +$5 0

0 5#$5

&

, M =

)

12(1#

$5)i 1

2(1 +$5)i

1 1

*

.

73

74 CHAPTER 9

For Question 9:

D =

;

A

A

<

2 0 0 00 #2 0 00 0 3 00 0 0 5

=

B

B

>

, M =

;

A

A

<

1 1 1 250 1 1 #30 0 5 210 0 0 14

=

B

B

>

.

For Question 10:

a) D =

;

<

#1 0 00 4 00 0 6

=

> , M =

;

<

#3 1 02 1 00 0 1

=

> .

b) D =

;

<

2 0 00 #3 00 0 3

=

> , M =

;

<

0 0 1#1 #1 06 1 0

=

> .

15. If v is an eigenvector of T then the coordinate vector [v]B of v with respect to the basis Bis the corresponding eigenvector for the matrix A.

16.1$2

%

1 11 #1

&

.

17.

;

A

A

A

A

A

A

<

#2$5

1$30

#1$6

05$30

1$6

1$5

2$30

#2$6

=

B

B

B

B

B

B

>

.

18. A5 =

%

#78 33055 #133

&

.

19. a) 6, t

%

12

&

, t %= 0, t " R; #4,

%

#34

&

, t %= 0, t " R.

b) P =

%

1 #32 4

&

, D =

%

6 00 #4

&

. c)

%

6n (#3) (#4)n

2! 6n 4(#4)n

&

.

20. When A is diagonalisable, Ak = MDkM!1 and x(k) = Akx(0). As a check, if you put k = 0in the answers below you should get A0 = I, whereas if you put k = 1 you should get A1 = A.

For Question 7:

a) Ak =1

2

%

3k + (#1)k 3k + (#1)k+1

3k + (#1)k+1 3k + (#1)k

&

.

b) The matrix is not diagonalisable.

c) Ak =1

9

%

3k+2 5(3k # (#6)k)0 9(#6)k

&

.

74

ANSWERS 75

d) Let &1 = 1 + i and &2 = 1# i. Then

Ak =

)

(12 +12 i)&

k1 + (12 # 1

2 i)&k2 i(&k1 # &k2)

#12 i(&

k1 # &k2) (12 # 1

2 i)&k1 + (12 +

12 i)&

k2

*

e) Let &1 = 5 +$3i and &2 = 5#

$3i. Then

Ak =1$3

)

(%32 + 1

2 i)&k1 + (

%32 # 1

2 i)&k2 &k1 # &k2

&k1 # &k2 (%32 # 1

2 i)&k1 + (

%32 + 1

2 i)&k2

*

.

f) Let &1 = 5 +$5 and &2 = 5#

$5. Then

Ak =i$5

)

12 i(1#

$5)&k1 # 1

2 i(1 +$5)&k2 #&k1 + &k2

&k1 # &k2 #12 i(1 +

$5)&k1 +

12 i(1#

$5)&k2

*

.

For Question 9:

Ak =

;

A

A

A

A

A

<

2k #2k + (#2)k #15(#2)k + 1

5(3k) #2(2k) + 18

35 (#2)k # 310 (3

k) + 2514(5

k)

0 (#2)k #15(#2)k + 1

5(3k) 18

35 (#2)k # 310 (3

k)# 314(5

k)

0 0 3k #32(3

k) + 32(5

k)

0 0 0 5k

=

B

B

B

B

B

>

.

For Question 10:

a) Ak =1

5

;

<

3(#1)k + 2(4)k 3(#1)k+1 + 3(4)k 02(#1)k+1 + 2(4)k 2(#1)k + 3(4)k 0

0 0 5(6)k

=

> .

b) Ak =1

5

;

<

5(3)k 0 00 #(2)k + 6(#3)k #(2)k + (#3)k

0 6(2)k # 6(#3)k 6(2)k # (#3)k

=

> .

21. "1, "2, "3 and "4 are arbitrary real numbers.

For Question 5:

a) y(t) = "1e3t%

23

&

+ "2e2t%

12

&

.

b) y(t) = "1et%

13

&

+ "2e!3t

%

11

&

.

For Question 9:

y(t) = "1e2t

;

A

A

<

1000

=

B

B

>

+ "2e!2t

;

A

A

<

1100

=

B

B

>

+ "3e3t

;

A

A

<

1150

=

B

B

>

+ "4e5t

;

A

A

<

25#32114

=

B

B

>

.

75

76 CHAPTER 9

For Question 10:

a) y(t) = "1e!t

;

A

<

#32

10

=

B

>+ "2e4t

;

<

110

=

> + "3e6t

;

<

001

=

> .

b) y(t) = "1e2t

;

A

A

<

0

#16

1

=

B

B

>

+ "2e!3t

;

<

0#11

=

> + "3e3t

;

<

100

=

> .

22. e2t%

"1

%

cos t# sin t

&

+ "2

%

sin tcos t

&&

23. a) 1, &

%

3#1

&

, & %= 0; 5, &

%

11

&

, & %= 0. b)

?

x1 = 3"et + )e5t,

x2 = #"et + )e5t.

24. a) x = 300et#200e3t, y = 150et#50e3t. b) x = #500+600e!2t, y = #100+200e!2t.

25. The solutions by the two methods are:

a) y(t) =

%

y(t)y(t)

&

= "1et%

11

&

+ "2et/5

)

1

15

*

, (matrix method)

y(t) = "1et + "2et/5. (calculus method)

b) y(t) =

%

y(t)y(t)

&

= "1e4t%

14

&

+ "2e!4t

%

1#4

&

, (matrix method)

y(t) = "1e4t + "2e!4t. (calculus method)

26. The matrix method given in notes is not applicable as the matrix is not diagonalisable.

27. a) A =

%

0 1# c

a # ba

&

.

28. a)500 years 0.9048: 0.0928: 0.00241000 years 0.8187: 0.1722: 0.0091

1000000 years 1.4! 10!87: 7! 10!44: 1

b) The associated matrix is not diagonalisable.

29. x(k) ' 1

3

%

21

&

30. In the 12th:

;

<

0.98 0.02 0.030.01 0.96 0.030.01 0.02 0.94

=

>

11 ;

<

300300300

=

> 2

;

<

378293229

=

> ,

76

ANSWERS 77

In the 24th:

;

<

0.98 0.02 0.030.01 0.96 0.030.01 0.02 0.94

=

>

23 ;

<

300300300

=

> 2

;

<

426280194

=

> .

31. In the 12th:

;

<

339262205

=

> total = 806; In the 24th:

;

<

339222154

=

> total = 715.

32. The population settles to the proportions 1.156 : 1.124 : 1.116 : 1.086 : 1 but eventually diesout.

77

78 CHAPTER 9

78

PAST CLASS TESTS

In the years up to 2007 there were 3 algebra class tests per session. From semester 2 2008 there willbe only 2 algebra class tests per semester so the pre-2008 tests included here do not have the samecoverage of material as the class tests for 2008 and onwards. The 2014 Information booklet forMATH1251 lists the material available for examination in the current schedule of class tests, as doespage (??) of these notes. Also there have been some changes to the syllabus for 2008 and beyondand some parts of the questions in the following pre-2008 class tests are no longer examinable.Thus the following pre-2008 tests should only be taken as a guide to the level of di!culty to beexpected in class test questions for 2008 and onwards.

Sample class tests from 2008 and onwards are included after all the pre-2008 tests and thesecorrespond to the current course syllabus and class test schedule. However, the content of the 2014class tests is defined in the current 2014 Information booklet for MATH1251.

79

UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS AND STATISTICS

MATH1251 MATHEMATICS FOR ACTUARIAL STUDIESAND FINANCE 1B Algebra S2 2007

TEST 1 VERSION 1B

This sheet must be filled in and stapled to the front of your answers

Student’s Family Name Initials Student Number

Tutorial Code Tutor’s Name Mark

Note: The use of a calculator is NOT permitted in this test

QUESTIONS (Time allowed: 20 minutes)

1. (3 marks)

Find the real and imaginary parts of (#$3 + i)200.

2. (4 marks)

Factorise z5 + 100 000 into real linear and quadratic factors.

3. (3 marks)

Prove that if arg z = # then(z/z)2 # (z/z)2 = 2i sin 4# .

Be sure to set out your argument clearly and logically.




TEST 2 VERSION 1A






1. (4 marks)

Let +a and +b be two fixed vectors in R3; as usual, denote by +u ·+v the dot product of two vectorsin R3. Prove that

S = {+x " R3 | +x · +a = +x ·+b }

is a subspace of R3. Note. To obtain full marks for this question you must give a clearlywritten, carefully explained and logically precise answer.

2. (1 mark)

Show thatS = {+x " R2 | x1 = x22 }

is not a subspace of R2.

3. (2 marks)

Let

A1 =

%

1 3#2 1

&

, A2 =

%

#1 #33 0

&

and A3 =

%

2 #1#5 1

&

.

Is {A1, A2, A3 } an independent set? Is it a spanning set for M2,2? Give reasons for youranswers.

4. (3 marks)

Prove thatB = { 1 + t, t+ t2, t2 + t3 }

is a basis forV = { p " P3 | p(#1) = 0 } .




TEST 3 VERSION 2B






1. (3 marks)

Prove that the map T : P3 ' P6 defined by

T (p(x)) = p(x2)

is a linear transformation. Note. To obtain full marks for this question you must give aclearly written, carefully explained and logically precise answer.

2. (1 mark)

Show that the map T : M2,2 ' R defined by

T

%

a bc d

&

= ad# bc

is not a linear transformation.

3. (3 marks)

Find a basis for the image of the matrix

A =

;

<

1 #2 #1 32 #4 3 7#1 2 #9 #5

=

> ,

and a basis for R3 containing this first basis.

4. (3 marks)

Let T : R2 ' R3 be a linear transformation; suppose that +a,+b are vectors in R2 such thatT (+a) = (1, 0, 0) and T (+b) = (0, 1, 0). Prove that the equation T (+x) = (0, 0, 1) has no solution.



MATH1251 MATHEMATICS FOR ACTUARIAL STUDIESAND FINANCE 1B ALGEBRA S2 2008

TEST 1 VERSION 2A






1. (3 marks)

Solve the equation z2 # (4 # i)z + (5 + i) = 0.

2. (3 marks)

Find all the fourth roots of 8 + 8$3 i. You may leave your answers in polar form.

3. (2 marks)

Draw a neat and accurate sketch of the region of the complex plane defined by

0 . Arg"

(z # i)3#

. !

2.

4. (2 marks)

Are the solutions of the following stable or unstable? Give reasons.

(i) The discrete time system 5xn+1 # 2xn + 4xn!1 = 0.

(ii) The continuous time system 5d2x

dt2# 2

dx

dt+ 4x = 0.




TEST 1 VERSION 2A






1. (1 mark)

Find a real number a such that

3'1 + 2i

a+ 3i

(

= 0 .

2. (3 marks)

Find the real and imaginary parts of (#1 + i)77.

3. (2 marks)

Does the discrete time system

5xn+1 # 19xn # 22xn!1 + 4xn!2 = 0

have unstable solutions? Give reasons for your answer.

4. (4 marks)

Factorise z6 + 1 into real linear and quadratic factors.




TEST 2 VERSION 3B






1. (2 marks)

Let +u,+v and +w be elements of a vector space V . Prove that if +w " span{+u,+v} then span{+u,+v, +w} =span{+u,+v}. Be sure to set out your argument clearly and logically.

2. (4 marks)

(i) Find a basis for the column space of the matrix

A =

;

<

1 2 #1 1#2 #1 3 03 9 #2 5

=

> .

(ii) Find a basis for R3 which includes the basis you found in (a).

3. (2 marks)

Let

+v1 =

;

<

1#32

=

> , +v2 =

;

<

#122

=

> , +v3 =

;

<

3#75

=

> and +w =

;

<

#225

=

> .

Find the coordinate vector of +w with respect to the ordered basis {+v1,+v2,+v3} for R3.

4. (2 marks)

Let A be a fixed 2!3 matrix and B a fixed 4!5 matrix. Prove that the function T : M3,4 'M2,5 defined by

T (X) = AXB

is a linear transformation.




TEST 2 VERSION 1A






1. (4 marks)

Let p1(t) = 1# 3t+ 2t2, p2(t) = #1 + 4t# 3t2, p3(t) = 2# 5t+ 3t2 and p4(t) = 1# t# 3t2.

(i) Is {p1, p2, p3, p4} a linearly independent set? Give reasons.

(ii) Is {p1, p2, p3, p4} a spanning set for P2? Give reasons.

2. (2 marks)

Let

+v1 =

;

<

1#23

=

> , +v2 =

;

<

2#37

=

> , +v3 =

;

<

#152

=

> and +w =

;

<

2#53

=

> .

Find the coordinate vector of +w with respect to the ordered basis {+v1,+v2,+v3} for R3.

3. (2 marks)

Let +a be a fixed non-zero vector in R3. Prove that the function T : R3 ' R3 defined by

T (+x) =+a · +x+a · +a

+a

is a linear transformation.

4. (2 marks)

Let T : V ' W be a linear transformation. Define the kernel of T and prove from thedefinition that the kernel is closed under addition. (You may not use the fact that thekernel is a vector space.)

Documents

MATH1251 Mathematics for Actuarial Studies and Finance …web.maths.unsw.edu.au/~potapov/1251_2014/math1251AlgProblems.pdf · Chapter 6 COMPLEX NUMBERS 6.1 A review of number systems