ANSWERS TO EXERCISES
ANSWERS TO EXERCISES
1.5 Exercises (page 7)
1.10 Exercises (page 13)
1. Yes; 2 5. Yes; 1 9. Yes; 1 13. Yes;
2. Yes; 2 6. No
10. Yes; 1 14. Yes;
3. Yes; 2 7. No
11. Yes;
15. Yes;
4. Yes; 2 8. No
12. Yes;
16. Yes;
17. Yes; dim = 1 +
if is even,
+ 1) if is odd
18. Yes; dim = if
+ 1) if is odd
19. Yes; k+1
23. (a) If a 0 and 0, set is independent, dim = 3 if one of a or is zero, set is de- pendent, dim = 2
independent, dim = 3 ; if a = dependent, dim = 2
(b) Independent, dim = 2
(c) If a
(e) Dependent; dim = 2 (f) Independent; dim = 2
(d) Independent; dim = 3
(h) Dependent; dim = 2 (i) Independent; dim = 2
(g) Independent; dim = 2
(j) Independent; dim = 2
1.13 Exercises (page 20)
10. (b) aarbitrary 6n
11. (c) 43 (d) g(t) =
a arbitrary
12. (a) No
13. (c)
(d)
Answers to exercises
1.17 Exercises (page 30)
2.4 Exercises (page 35)
1. Linear; nullity 0, rank 2
13. Nonlinear
2. Linear; nullity 0, rank 2
14. Linear; nullity 0, rank 2
3. Linear; nullity 1, rank 1
15. Nonlinear
4. Linear; nullity 1, rank 1
16. Linear; nullity 0, rank 3
5. Nonlinear
17. Linear; nullity 1, rank 2
6. Nonlinear
18. Linear; nullity 0, rank 3
9. Linear; nullity 0, rank 2
21. Nonlinear
Linear nullity 0, rank 2
22. Nonlinear
11. Linear ; nullity 0, rank 2
23. Linear ; nullity 1, rank 2
12. Linear ; nullity 0, rank 2
24. Linear; nullity 0, rank n+ 1
25. Linear; nullity rank infinite
26. Linear; nullity infinite, rank 2
27. Linear; nullity 2, rank infinite
28. N(T) is the set of constant sequences; T(V) is the set of sequences with limit 0
29. (d)
1, cos x, sin x} is a basis for T(V); dim T(V) = 3 (e) N(T) = S =
with c 0, th =0, then c=
are not both zero but otherwise arbitrary; if
where is
but otherwise arbitrary.
2.8 Exercises (page 42)
3. Yes; y
10. Yes;
y=v-1
4. Yes; y=
8. Yes; y
15. Yes;
9. No
16. Yes;
17. Yes;
18. Yes;
19. Yes;
y-v-u,
20. Yes;
25. (S + = + ST + TS +
Answers to exercises
26. (a) z) = + + z, x + x) z) = (z, z + z + + x) ; (ST TWX, y, =
x z,
y, = (x, ;
=(x,2x y, = (3x +
= 3 2x + 12x2;
(b) p(x) = ax, a an arbitrary scalar (c) p(x) =
= 8 192x
a and b arbitrary scalars
R(V) = {p p is constant} N(S) = {p p is constant} ; S(V) = V; N(T) =
= 0} (c) = s (d)
T(V) = {p
=Z R;
32. T is not one-to-one on V because it maps all constant sequences onto the same sequence
2.12 Exercises (page 50)
= 0 ifj k (b) The zero matrix 0 =
1. (a) The identity matrix Z =
where
= 1 ifj = k, and
where each entry
The matrix
where
is the identity matrix of part (a)
3 i + 4j + 4k ; nullity 0, rank 3
1 -3
-1 -5
Answers to exercises
7. (a) j+ = (0, -2) nullity 1, rank 2
1 -1
8. (a) (5, 0, -1); nullity 0, rank 2
9. (a) (-1, -3, -1); nullity 0, rank 2
10. (a) ; nullity 0, rank 2
(b)
4 (c) a = 5, = 4
Answers to exercises
0 000 1 0 0 0 0 0 -48
20. Choose x, 1) as a basis for and x) as a basis for W. Then the matrix of TD is
2.16 Exercises (page
AC =
7. A”
Answers to exercises
where b and c are arbitrary, and a is any solution of the equation
= - bc
where a is arbitrary
where b and c are arbitrary and a is any solution of the equation
For those which commute
Exercises (page 67) =
2. No solution
13. -1 0 1
0 0 l-2
0 0 -1
0 0 0 1 0 -1
16. 9 0 -3
Answers to exercises
2.21 Miscellaneous exercises on matrices (page 68)
3. P=
where b and c are arbitrary and a is any solution
of the quadratic equation
3.6 Exercises (page 79)
3. (b) (b a)(c a)(c
b)(a + b + c) and (b
a)(c
a)(c
b)(ab + ac + bc)
4. (a) 8
(b) (b a)(c
f3 fl
f3
fl
8. (b) If F =
then
10 det A=
16, det
3.11 Exercises (page 85)
6. det A= (det
7. (a) Independent
(b) Independent
(c) Dependent
3.17 Exercises (page 94)
Answers to exercises
3. (a) I=-3
4.4 Exercises (page 101)
5. Eigenfunctions: = Ct”, where 0
6. The constant polynomials
7. Eigenfunctions : =
, where C 0
8. Eigenfunctions: =
where 0
Eigenvectors belonging to = 0 are all constant belonging to = -1 are
nonconstant sequences with limit 0
4.8 Exercises (page 107)
Eigenvectors
dim E(1)
1. (a) 1,1
Answers to exercises
3. If the field of scalars is the set of real numbers R, then real eigenvalues exist only when sin = 0, in which case there are two equal eigenvalues, I, = I, = cos where cos = 1 or
1. In this case every
vector is an eigenvector, so dim
= dim = 2.
If the field of scalars is the set of complex numbers C, then the eigenvalues are cos + sin I, = cos
i sin If sin = 0 these are real and equal. If sin 0 they are distinct complex conjugates; the eigenvectors belonging to I, are
0 ; those be- longing to are
4. where b and c are arbitrary and a is any solution of the equation
1 real and equal if A = complex conjugates if A < 0.
5. Let A =
, and let A = (a
+ 4bc. The eigenvalues are real and distinct if A > 0,
dim E(I)
8. 1, 1, -1, -1 for each matrix
4.10 Exercises (page 112)
2. (a) Eigenvalues 1, 3; C =
, where
cd 0
(b) Eigenvalues 6,
(c) Eigenvalues 3, 3; if a nonsingular C exists then
so AC = A=
(d) Eigenvalues 1; if a nonsingular C exists then
so AC = C, A = Z 3.
4. (a) Eigenvalues 1, 1, - 1 ; eigenvectors (1, 0, (0, 1, 0), (1, 0, - 1 ) ;
c=o 1 1 0 -1 0 (b) Eigenvalues 2, 2, 1; eigenvectors (1, 0,
(1, -1, 1);
c= -1 -1 l-l 1
Answers to exercises
5. (a) Eigenvalues 2, 2; eigenvectors
(b) Eigenvalues 3, 3; eigenvectors
0. If C =
, then
6. Eigenvalues 1, 1, 1; eigenvectors
Chapter 5
5.5 Exercises (page 118)
3. (b) is Hermitian if is even, skew-Hermitian if is odd
7. (a) Symmetric (b) Neither
(c) Symmetric
(d) Symmetric
5.11 Exercises (page 124)
1. (a) Symmetric and Hermitian (b) None of the four types (c) Skew-symmetric (d) Skew-symmetric and skew-Hermitian
5. Eigenvalues = 0, = 25; orthonormal eigenvectors
= -2i; orthonormal eigenvectors
= -4; orthonormal eigenvectors
(3, -5, -1).
632 Answers to exercises
8. Eigenvalues = 1,
= -4; orthonormal eigenvectors =
4, = (5, -3, -4).
34 -3 -4
9. (a), (b), (c) are unitary; (b), are orthogonal
11. (a) Eigenvalues =
orthonormal eigenvectors
-i).
5.15 Exercises (page 134)
1. (a) A =
(b)
0 (b)
2. (a) A =
11 1 (d) C=
3. (a) A =
(b) =
A ,=
(c) = +
where t = l/d4 + 2
(d) C = t
where t =
34 -12
4. (a) A =
(b)
(b)
A, =
Answers to exercises
1 14. Ellipse; center at
(d) C=
-3 0 -1 4
8. Ellipse; center at
9. Hyperbola; center at
15. Parabola; vertex at
10. Parabola; vertex at
16. Ellipse; center at ( 1,
11. Ellipse; center at
17. Hyperbola; center at
12. Ellipse; center at (6, -4)
18. Hyperbola; center at (
13. Parabola*, vertex at
19. -14
5.20 Exercises (page 141)
8. a =
Chapter 6
6.3 Exercises (page 144)
4. Four times the initial amount
9. y =
cos 2x + sin 2x)
5. = or
10. y =
Answers to exercises
= -12y =
6.9 Exercises (page 154)
13. f(x) cos mx
6.15 Exercises (page 166)
1. = -2x
3. = (x
7. =x
4. = sin x
9. = sin 2x +
cos 2x
Yl
PA
15. (b)
(c)
(d)
16. y= + +
dx
17. y = (A +
sin 2x + (B + log
cos 2x
18. y =
19. y = (A + +
dx +
dx
Answers to exercises
6.16 Miscellaneous exercises on linear differential equations (page 167)
1. u(x) = ;
2. u(x) = sin 5x
sin 3x
3. u(x) = Q(x) =
6.21 Exercises (page 177)
(b) where = 1 or -2, and x = +
6.24 Exercises (page 188)
+ sin x
Answers to exercises
9. (a) =
(c) =
+ provided that + 2) is not an integer; otherwise replace the appropriate by K
7.4 Exercises (page 195)
3. (b) =
7.12 Exercises (page 205)
= (cosht)Z + (sinht)A =
Answers to exercises
4. (a) = A, = 1+
= + (sinh t)A =
8. (a) =0 if
9. (a) =A if
11. =Z+ +
7.15 Exercises @age 211)
t)A
3. = 2t + 2)Z +
7. (b) =
+ (6t
Answers to exercises
8. = t+ t
7.17 Exercises (page 215) 2. (Cj
= ‘This gives the particular solution
B, where B=
7.20 Exercises (page 221)
4. (c:) Y(x) =
5. If A(x) = then Y(x) = B+
where + +
for k
7.24 Exercises (page 230)
1. (a) Y(x) = =
if is odd;
if is even
Answers to exercises
2. Y,(x) = + + +
3. Y,(x) = + + + 4x’
(b) (c) Y(x) = 1 + x + +
(d) Y(x) = tan = +
10. (d) Y,(x) = 0; Y,(x) = 0 if
0 if
Y,(x) =
if
0 if
(f) Y,(x) = lim
; lim Y,(x) =
8.3 Exercises (page 245)
2. All open except (d), (e), (h), and (j)
3. All open except (d)
5. (e) One example is the collection of all 2-balls
l/k), where k=
6. (a) Both (b) Both
(e) Closed (f) Neither (g) Closed
(c) Closed
(h) Neither
(i) Closed
(j) Closed
(k) Neither (1) Closed
8. (e) One example is the collection of all sets of the form
1 l/k} for
k=
3, . . . . Their union is the open ball
10. No
640 Answers to exercises 8.5 Exercises (page 251)
withy 0 (d) (e) All (x, with 0
All (x, y) with xy 1
(j)
5. lim y) does not exist if 0
; NO 7. y =
f not continuous at
8.9 Exercises (page 255) 1. =
(b) All points on the line 2x + 3y = (c) All points on the plane +
= 2y sin (xy) + cos
5. = + ; = y/(x2 +
af
= =2x/(x 8. =
where a = (a,, . . . ,
Answers to exercises
17. af
18. n =
22. (b) One example is f (x) = . , where is a fixed
3. (1, 0), in the direction of i; (-1, 0), in the direction of
4. 2i + 2j ;
5. (a, b, c) = -8) or (-6, -24, 8)
6. The set of points (x, y) on the line 5x
11. (b) implies (a) and (c); (d) implies (a), (b), and (c); (f) implies (a)
8.17 Exercises (page 268)
1. (b) F”(t) = +2 X’(t) Y’(t) + + X”(t) + Y”(t)
2. (a) F’(t) = + 2t ; F”(t) =
sin(cos t ; F”(t) (5
(b) F’(t) = (2 t + (3
2 sin
cos (cos t + (14
t 4 cos
sin (cos t t) exp
where exp (u) =
F”(t) = + exp
+ exp
Answers to exercises
3. (a) (b)
(a) (1 + +
or any scalar multiple thereof (b) cos =
cos [cos +
af af
af af
af af af af
af af
-r cos sin + r
- + r cos sin
-sin + cos -sin + cos
(b) y, z) = + + R(z)k, where P, Q, R are any three functions satisfying
0 -6cos9
2x 1 1
15. (a)
y, z) =
1 w) =
0 cos
sin v
v, w) =
sin v + 2u
sin v +
0, w) =
Answers to exercises
8.24 Miscellaneous exercises (page 281)
1. One example: y) = 3x when x = y) = 0 otherwise ;
= -1;
0) does not exist
3. (a) =0 (b) Not continuous at the origin
+ 3 Y’(f) + 3
Y’(r)]2
Y’(t) +
assuming the mixed partial derivatives are independent of the order of differentiation
10. (a) = A’(t)
f [A(t),
+ B’(t)
f [x, B(t)]
(b) =
13. A sphere with center at the origin and radius
14. f(x) =
Chapter 9
9.3 Exercises (page 286)
1. f(x,
yj 6. =x-y
D = -3;
9.8 Exercises (page 302)
(24i
j+
k)
Answers to exercises
5. 2i + + k , or any
scalar multiple thereof
10. = + + sin (y +
= f [x +
=f
+ g(y)]
+f
9.13 Exercises (page 313)
1. Absolute minimum at (0, 1)
2. Saddle point at (0, 1)
3. Saddle point at
4. Absolute minimum at each point of the line y = x + 1
5. Saddle point at (1, 1)
6. Absolute minimum at
7. Saddle point at
8. Saddle points at and at (x, 0), all x; relative minima at (0, y), 0 < y < 6 ; relative maxima at
and at (0, y) for y < 0 and y > 6
9. Saddle point at
relative minimum at (1, 1)
10. Saddle points at
+ a/2, 0) , where is any integer
11. Absolute minimum at
saddle point at
12. Absolute minimum at
absolute maximum at (1, 3)
13. Absolute maximum at
relative maximum at relative minimum at
absolute minimum at
saddle points at (0, and
14. Saddle point at (1, 1)
15. Absolute maximum at each point of the circle
= 1 ; absolute minimum at
17. (c) Relative maximum at (2, 2); no relative minima; saddle points at (0, and (3, 3)
at and saddle points at (0, 0), ( fl, 0), and (0,
18. Relative maximum at
and
relative minimum
absolute maximum 1 at (1, -1) and (-1, 1); absolute minimum 1 at (1, 1) and ( 1, 1)
y* =
-x*.Thena =
andb -ax*
22. = , and let
, where the sums are for i =
. . . , n. Then
Answers to exercises
-ax*
25. Eigenvalues 4, 16, 16; relative minimum at (1, 1, 1)
9.15 Exercises (page 318)
1. Maximum value is
no minimum
2. Maximum is 2; minimum is 1
3. (a) Maximum is
at
minimum is
ab
ab (
(b) Minimum is
at
no maximum
4. Maximum is I +
where is any integer; minimum is 1
at the points
at +
where is any integer
5. Maximum is 3 at
minimum is -3 at
6. 1) and (O,O, -1)
9. (a + b +
a+b+c’a+b+c’a+b+c
14. Angle is ; width across the bottom is c/3 ; maximum area is
Chapter 10
10.5 Exercises (page 328)
2. 9. -3x-
5. 0 (a) -2
Answers to exercises
10.9 Exercises (page 331)
4. 0 12. moment of inertia =
10.13 Exercises (page 336)
1. All except (f) are connected
6. (a) Not conservative (b)
7. (b) 3
10. + 4 ; minimum occurs when b=
10.18 Exercises (page 345)
8. is not a gradient
9. f is not a gradient
10. f is not a gradient
+Cifn = -1
15. = + Cifp
= logr + Cifp = -2
16. +C
10.20 Exercises (page 349)
2. =c
Answers to exercises
3. -y/2 + (sin
is an integrating factor
(b) + sin y = C ;
cos y is an integrating factor
10. + = c,
= C, respectively;
is a common integrating factor
Chapter 11
11.9 Exercises (page 362)
11.15 Exercises (page 371)
2. + cos 1 + sin 1 cos 2 2 sin 2
3. e 7.
4. 8. (a)
(c) 9.
Answers to exercises
dy ] dx =
20. y =0, y =xtanc,
(b) +
22. n=l
11.18 Exercises (page 377)
10. assuming the x- and y-axes are chosen along sides
11.22 Exercises (page 385)
1. (a) -4
Answers to exercises
Exercises (page 391)
1. (b) 0
3. As many as three
4. As many as seven
5. (a) -3 6.
11.28 Exercises (page 399) cos r sin
+ sin
1 + sin
where g(0) =
14. (a)
17. (a) 1+
651 18. (a)
Answers to exercises
if p = 1 . r) tends to a finite limit whenp > 1
11.34 Exercises (page 413)
20. 22. On the axis at distance
from the base
23. On the axis at distance
from the base
24. On the axis of symmetry at distance . from the “cutting plane” of the hemispheres 25. = = =
(assuming the specified corner is at the origin)
12.4 Exercises (page 424)
2. + = z;
cos vi
sin vj
ubuk
Answers to exercises
12.6 Exercises (page 429)
5. (a) A circular paraboloid (b)
11. (a) A unit circle in the xy-plane; a unit semicircle in the xz-plane, with z
0 ; a unit semicircle in the plane x = with z 0 (b) The hemisphere
The sphere + + z = 1 except for the North Pole; the line joining the North Pole and (x, y, intersects the xy-plane at (u, v, 0)
12.10 Exercises (page 436)
9. On the axis of the cone, at a distance
from the center of the sphere
cos
cos
Answers to exercises
12.13 Exercises (page 442)
1. 0 3. -4
12.15 Exercises (page 447)
sin y ; curl
y, z) = + j
div y, z) =
x sin (xy)
sin
curl y, z) = sin
+ y sin
(e) div z) = 2x sin y
sin (xz) xy sin z cos (cos z)
curl y, = sin (cos
+ cos (xz) cos
cos
y sin (cos
4. n=-3
5. No such vector field
10. One such field is
y, z) =
11. x r) =O; curl
r) = (c +
13. + Exercises (page 452)
1. (3x 2z)j xk is one such field
xz)k is one such field
y) for somefindependent of z
5. y, z)
j satisfies curl G =
at all points not on the z-axis
6. f(r) =
12.21 Exercises (page 462)
2. (a) 14477
(c)
Chapter 13
13.4 (page 472)
A ;n
3. (ii)
(iii)
(a) A’ B’
A (A B’) u (A’ B)
A’ u B’
Answers to exercises
13.7 Exercises (page 477)
13.9 Exercises (page 479)
10. = P(A) + P(B)
13.11 Exercises (page 485)
3: 54 4. {H,
x {H, x
24 outcomes
Answers to exercises
11 . 72 = 123552 (not including triplets or quadruplets)
(b) 5148 (c) 36 (not including
13.14 Exercises (page 490)
2. (a) P(A)
15. (a) P(A) = P(B) = P(C) = ;
C) 0
13.18 Exercises (page 499)
(b) Yes
(d) and
and ,
and
and
2. (a)
Answers to exercises
7. It is advantageous to bet even money
13.20 Exercises (page 504)
f(k) = 2k f(k) = 3”
f(k) =
is the kth prime
one such function is f(k) = (g(k), h(k)), where
+ m(k) =
and
m(k) = 2
where denotes the greatest integer
(e) f(k) =
and h(k) are as defined in part (d)
13.22 Exercises (page 507)
1. max=l,
13.23 Miscellaneous exercises on probability (page
8. (a) (b) 6
I --
4. (a)
Answers to exercises
Chapter 14
14.4 Exercises (page 513)
14.8 Exercises (page 523)
3. (b) -2) =
5. (a) c = (c) (d) No such
p(t) = 0 for t
F(t) = on F(t) =
= 0 for < 0, F(t) =
on
F(t) =
on
on
F(t) = 1 for t 4
denotes the greatest
integer = 0 for < 0;
= 1 for >
9. p(k) = ,k = 0 for t 0,
10. (a) at = -1 and =
= 0 elsewhere
= 0 for t < -1 ;
= for - 1
t<1;
= 1 for t
Answers to exercises
14.12 Exercises (page 532)
1. (a) c = 1 ; f(t) = 1 if 0
1 ; f(t) = 0 otherwise
< 0 ; F(t) =
if 0
2; F(t) = 1 i f
5. (a) ift (b)
+ 7t 2 if < F(t) = 1 if
F(t) =
t if < ; F(t) =
(b) Let each Styx train arrive minutes before a
10. F(t) = +
14.16 Exercises (page 540)
1. (a) 105 (b) 10.05
2. (a) 1)
3. (a) 1 (b)
4. F(t) = 0 if < c ; F(t) = 1
if c
9. (a) 0.5000 (b) 0.1359 (c) 0.9974 (d) 0.0456
10. (a) 0.675 (b) 0.025
11. (a) 0.6750 (b)
Answers to exercises
12. (a) 0.8185 (b) 0.8400
13. 75.98 inches
14. mean = b, variance = = Cl if < 0 ;
= 0 if 0 ;
if 0
14.18 Exercises (page 542)
Let be the inverse of
defined on the open interval (a,
Then = 0 if
= 1 if b ; if a < < ;
if a < b ;
14.22 Exercises (page 548)
(b)
3. (a) F(x,y) =
ifx>b and
1 if x
and y >
y) = 0 otherwise
if
if x < a
(c) and Y are independent
= 0 if
and Y are not independent
10. + a, +
14.24 Exercises (page 553)
and are not independent
Answers to exercises
2. (b) = 2 if 0 1 ;
= 0 otherwise
and V are independent
3. (b) = if > 0,
v) = 0 otherwise =
<O; -exp
exp
14.27 Exercises (page 560)
7. (a) E(X) = Var (X) = (b) None (c) E(X) =
Var (X) = (d) E(X) = m, Var (X) =
8. (a) C(r) =
(d) X has a finite expectation when r> ; E(X) Cl
Variance is finite for r > 3 ; Var (X) =
9. = E(Y) =
; E(Z) =
E(X +
+ Z) =
10. E(X)+
12. (a) (b)
14.31 Exercises (page
6. Chebyshev’s inequality gives
tables give 0.0027
8. (b) 0.6826
9. (b) 0.0796
10. (a) 0.0090 (b) 0.0179
Chapter 15
15.5 Exercises (page 577)
2. (a) No
3. (a) Neither (b) Yes
(b)
(c) Yes
(c) (d) Neither
Yes Neither (h) Neither
Answers to exercises
8. (b) The polynomial in (a) plus
3/e, c =
7/e)
+ 14x2 (b)
(b) P(x) = (18 6e)x + 4e 10;
15.9 Exercises (page 585)
1. (a) P(x) =
+ 13x + 12)
(b) P(x) = 5x + 6) (c) P(x) =
5x
(d) P(x) = +
P(x) = + 10x 5 2. P(x) =
4. Q(x) = +
Q(x) = +
3x
P(32) = f(32)
P(32)
P(32) = ; f(32)
P(32) ; f(32)
for 1 the one and only polynomial P of degree
1 and let
= (x
satisfying the conditions
is given by P(x) =
16. x
Answers to exercises
15.13 Exercises (page 593)
4. (b) 8 -5040
22449 -4536 546 - 3 6 1 10 -3628800
118124 -67284
723680 -269325 63273 -9450 870 - 4 5 1 (c)
5880 750 45 1 (d) -1+6x
15.18 Exercises (page 600) 2. (b)
7. Q(x) =
15.21 Exercises (page 610)
1. (a) 0.693773 where 0.000208 0.001667. This gives the inequalities 0.6921 < log2 < 0.6936 (b) = 578 2. (a) =
3. (a) c =
Answers to exercises
10. (a) log 2 = 0.693254 where 0.000016 0.000521 this leads to the inequalities
0.69273 < log 2 < 0.69324
(b) log 2 = 0.69315023 , where 0.00000041 0.00001334 this leads to the in- equalities 0.693136 < log 2 0.693149
11. (d) log 2 = 0.693750 where 0.000115 0.003704 this leads to the inequalities
0.69004 < log 2 < 0.69364