Covers All 12 Chapters
SOLUTIONS MANUAL
, TABLE OF CONTENTS
Page
Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations 86
Chapter 3. Vector Calculus 126
Chapter 4. Functions of a Complex Variable 194
Chapter 5. Laplace Transforms 270
Chapter 6. The z Transform 369
Chapter 7. Fourier Series 413
Chapter 8. The Fourier Transform 489
Chapter 9. Partial Differential Equations 512
Chapter 10. Optimization 573
Chapter 11. Applied Probability and Statistics 639
iii
, 1
Matrix Analysis
Exercises 1.3.3
1(a) Yes, as the three vectors are linearly independent and span three-
dimensional space.
1(b) No, since they are linearly dependent
⎡ ⎤ ⎡ ⎡ ⎤
3 1 1
⎤
⎣ 2 ⎦ − 2⎣ = ⎣ 2⎦
0⎦ 3
5 1
1(c) No, do not span three-dimensional space. Note, they are also linearly
dependent.
2 Transformation matrix is
= ⎡ 1 ⎤ ⎡ ⎤ ⎡ ⎤
1 −1 0 0 11
0 0
0 = √√
2
1
− √12 0
0
A √ 2⎣ √ ⎦ ⎣ ⎦ ⎣ 21 ⎦
12
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.
◯
c Pearson Education Limited 2011
, ◯
c Pearson Education Limited 2011
SOLUTIONS MANUAL
, TABLE OF CONTENTS
Page
Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations 86
Chapter 3. Vector Calculus 126
Chapter 4. Functions of a Complex Variable 194
Chapter 5. Laplace Transforms 270
Chapter 6. The z Transform 369
Chapter 7. Fourier Series 413
Chapter 8. The Fourier Transform 489
Chapter 9. Partial Differential Equations 512
Chapter 10. Optimization 573
Chapter 11. Applied Probability and Statistics 639
iii
, 1
Matrix Analysis
Exercises 1.3.3
1(a) Yes, as the three vectors are linearly independent and span three-
dimensional space.
1(b) No, since they are linearly dependent
⎡ ⎤ ⎡ ⎡ ⎤
3 1 1
⎤
⎣ 2 ⎦ − 2⎣ = ⎣ 2⎦
0⎦ 3
5 1
1(c) No, do not span three-dimensional space. Note, they are also linearly
dependent.
2 Transformation matrix is
= ⎡ 1 ⎤ ⎡ ⎤ ⎡ ⎤
1 −1 0 0 11
0 0
0 = √√
2
1
− √12 0
0
A √ 2⎣ √ ⎦ ⎣ ⎦ ⎣ 21 ⎦
12
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.
◯
c Pearson Education Limited 2011
, ◯
c Pearson Education Limited 2011
