Test Bank: Conv Solutions to Advanced Modern Engineering Mathematics by Burley, 2019 (5th Edition) - Graded A+ C...

STUDY NOTES & REFERENCE GUIDE




Covers All 12 Chapters




SOLUTIONS MANUAL




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TABLE OF CONTENTS
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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
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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




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1
Matrix Analysis ···




Exercises 1.3.3
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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⎣ 0⎦ = ⎣ 2 ⎦
5 1 3


1(c) No, do not span three-dimensional space. Note, they are also linearly
dependent.


2 Transformation matrix is
= ⎡1 ⎤⎡ ⎤ ⎡ ⎤
1 −1 0 10 10 0 = √√ 1
2
1
−√ 2 0
0
A √ 2⎣ √ ⎦⎣ ⎦ ⎣ 12 21 ⎦
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.



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2 Glyn James, Advanced Modern Engineering Mathematics, 4th Edition


3(b) No, does not span the required four-dimensional space. Thus a general
cubic cannot be written as a linear combination of

(1 − x), (1 + x), (1 − x3), (1 + x3)


as no term in x2 is present.



3(c) Yes as linearly independent set spanning the four-dimensional space


a(1 − x)+ b(1 + x) + c(x2 − x3) + d(x2 + x3)


= (a + b) + (b − a)x + (c + a)x2 + (d − c)x3

≡ α + βx + γx2 + δx3


3(d) Yes as a linear independent set spanning the four-dimensional space


a(x − x2) + b(x + x2) + c(1 − x3) + d(1 + x3)


= (a + b) + (b − a)x + (c + d)x2 + (d − c)x3

≡ α + βx + γx2 + δx3


3(e) No not linearly independent set as

(4x3 + 1) = (3x2 + 4x3) − (3x2 + 2x) + (1 + 2x)



4 x + 2x3, 2x − 3x5, x + x3 form a linearly independent set and form a basis
for all polynomials of the form α + βx3 + γx5 . Thus, S is the space of all odd
quadratic polynomials. It has dimension 3.



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