?
A?
VI
TU
_S
ED
OV
PR
AP
,AP
TABLE OF CONTENTS
Solutions Manual: Advanced Modern Engineering Mathematics, 4th Edition
PR By Glyn James
Chapter 1 Matrix Analysis
Chapter 2
OVNumerical Solution of Ordinary Differential Equations
Chapter 3 Vector Calculus
Chapter 4 Functions of a Complex Variable
Chapter 5
ED
Laplace Transforms
Chapter 6 The z Transform _S
Chapter 7 Fourier Series
Chapter 8 The Fourier Transform
Chapter 9 Partial Differential Equations
TU
Chapter 10 Optimization
Chapter 11 Applied Probability and Statistics VI
A?
?
,AP
TABLE OF CONTENTS
PR
Page
OV
Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations 86
Chapter 3. Vector Calculus 126
ED
Chapter 4. Functions of a Complex Variable 194
Chapter 5. Laplace Transforms 270
Chapter 6. The z Transform 369
Chapter 7. Fourier Series
_S 413
Chapter 8. The Fourier Transform 489
Chapter 9. Partial Differential Equations 512
TU
Chapter 10. Optimization 573
Chapter 11. Applied Probability and Statistics VI 639
A?
?
iii
, AP
PR
1
Matrix Analysis
OV
Exercises 1.3.3
1(a) Yes, as the three vectors are linearly independent and span three-
ED
dimensional space.
1(b) No, since they are linearly dependent
_S
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
3 1 1
⎣ 2 ⎦ − 2⎣ 0⎦ = ⎣ 2 ⎦
5 1 3
TU
1(c) No, do not span three-dimensional space. Note, they are also linearly
dependent.
VI
2 Transformation matrix is
⎡
1 1 0 ⎤ ⎡ 1 0 0 ⎤ ⎡ √1
A? ⎤
√1 0
= 1 1 −1 0 0 1 0 = √2 − √2 2 0
A √2 ⎣ √
⎦⎣ ⎦ ⎣ 12 1 ⎦
0 0 2 0 0 1 0 0 1
?
Rotates the (e1, e2) plane through π/4 radians about the e3 axis.
3 By checking axioms (a)–(h) on p. 10 it is readily shown that all cubics
ax3 + bx2 + cx + d form a vector space. Note that the space is four dimensional.
3(a) All cubics can be written in the form
ax3 + bx2 + cx + d
and {1, x, x2, x3} are a linearly independent set spanning four-dimensional space.
Thus, it is an appropriate basis.
A?
VI
TU
_S
ED
OV
PR
AP
,AP
TABLE OF CONTENTS
Solutions Manual: Advanced Modern Engineering Mathematics, 4th Edition
PR By Glyn James
Chapter 1 Matrix Analysis
Chapter 2
OVNumerical Solution of Ordinary Differential Equations
Chapter 3 Vector Calculus
Chapter 4 Functions of a Complex Variable
Chapter 5
ED
Laplace Transforms
Chapter 6 The z Transform _S
Chapter 7 Fourier Series
Chapter 8 The Fourier Transform
Chapter 9 Partial Differential Equations
TU
Chapter 10 Optimization
Chapter 11 Applied Probability and Statistics VI
A?
?
,AP
TABLE OF CONTENTS
PR
Page
OV
Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations 86
Chapter 3. Vector Calculus 126
ED
Chapter 4. Functions of a Complex Variable 194
Chapter 5. Laplace Transforms 270
Chapter 6. The z Transform 369
Chapter 7. Fourier Series
_S 413
Chapter 8. The Fourier Transform 489
Chapter 9. Partial Differential Equations 512
TU
Chapter 10. Optimization 573
Chapter 11. Applied Probability and Statistics VI 639
A?
?
iii
, AP
PR
1
Matrix Analysis
OV
Exercises 1.3.3
1(a) Yes, as the three vectors are linearly independent and span three-
ED
dimensional space.
1(b) No, since they are linearly dependent
_S
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
3 1 1
⎣ 2 ⎦ − 2⎣ 0⎦ = ⎣ 2 ⎦
5 1 3
TU
1(c) No, do not span three-dimensional space. Note, they are also linearly
dependent.
VI
2 Transformation matrix is
⎡
1 1 0 ⎤ ⎡ 1 0 0 ⎤ ⎡ √1
A? ⎤
√1 0
= 1 1 −1 0 0 1 0 = √2 − √2 2 0
A √2 ⎣ √
⎦⎣ ⎦ ⎣ 12 1 ⎦
0 0 2 0 0 1 0 0 1
?
Rotates the (e1, e2) plane through π/4 radians about the e3 axis.
3 By checking axioms (a)–(h) on p. 10 it is readily shown that all cubics
ax3 + bx2 + cx + d form a vector space. Note that the space is four dimensional.
3(a) All cubics can be written in the form
ax3 + bx2 + cx + d
and {1, x, x2, x3} are a linearly independent set spanning four-dimensional space.
Thus, it is an appropriate basis.
