, ADVANCED_MODERN_ENGINEERING MATHEMATICS 4TH Ed
TABLEOF CONTENTS
Page
Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations86
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
c
, 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 ⎤
1 1
Transformat⎤ io⎡ n matrix 0 0 1 0 0
is⎤ ⎡
1 ⎣ √1 1
⎦ 2 2 ⎦
1 1
⎦ ⎣ ⎣
A √ √
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
TABLEOF CONTENTS
Page
Chapter 1. Matrix Analysis 1
Chapter 2. Numerical Solution of Ordinary Differential Equations86
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
c
, 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 ⎤
1 1
Transformat⎤ io⎡ n matrix 0 0 1 0 0
is⎤ ⎡
1 ⎣ √1 1
⎦ 2 2 ⎦
1 1
⎦ ⎣ ⎣
A √ √
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
