barcode reader code in asp.net Linear Algebra and Complex Numbers in Software

Print QR Code ISO/IEC18004 in Software Linear Algebra and Complex Numbers

Appendix A
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Linear Algebra and Complex Numbers
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A1/n = (Aej )1/n = A1/n ej 1/n + k2 = n A n
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k = 0, 1, 2,
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(A20)
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EXAMPLE A3
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Perform the following operations given that A = 2 + j 3 and B = 5 j 4 (a) A + B (b) A B (c) 2A + 3B
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Solution
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A + B = (2 + 5) + j (3 + ( 4)) = 7 j A B = (2 5) + j (3 ( 4)) = 3 + j 7 For part c, 2A = 4 + j 6 and 3B = 15 j 12 Thus, 2A + 3B = (4 + 15) + j (6 + ( 12)) = 19 j 6
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EXAMPLE A4
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Perform the following operations both in rectangular and polar form, given that A = 3 + j 3 and B = 1 + j 3 (a) AB (b) A B
Solution
(a) In rectangular form:
AB = (3 + j 3)(1 + j 3) = 3 + j 3 3 + j 3 + j 2 3 3 = (3 + j 2 3 3) + j (3 + j 3 3) = (3 3 3) + j (3 + j 3 3)
To obtain the answer in polar form, we need to convert A and B to their polar forms: A = 3 2ej 45 = 3 2 45 B = 4ej 60 = 2 60 Then, AB = (3 2ej 45 )( 4ej 60 ) = 6 2 105
(b) To nd A B in rectangular form, we can multiply A and B by B A 3 + j3 1 j 3 = B 1+j 31 j 3 Then, (3 + 3 3) + j (3 3 3) A = B 4
Appendix A
Linear Algebra and Complex Numbers
In polar form, the same operation may be performed as follows: 3 2 45 3 2 3 2 A B= (45 60 ) = 15 = 2 60 2 2
Euler s Identity This formula extends the usual de nition of the exponential function to allow for complex numbers as arguments Euler s identity states that ej = cos + j sin (A21)
All the standard trigonometry formulas in the complex plane are direct consequences of Euler s identity The two important formulas are: cos = ej + e j 2 sin = ej e j 2j (A22)
EXAMPLE A5
Using Euler s formula, show that cos = ej + e j 2
Solution
Using Euler s formula ej = cos + j sin Extending the above formula, we can obtain e j = cos( ) + j sin( ) = cos j sin Thus, cos = ej + e j 2
Check Your Understanding
A4 In a certain AC circuit, V = ZI where Z = 775 90 and I = 2 45 Find V A5 In a certain AC circuit, V = ZI where Z = 5 82 and V = 30 45 Find I A6 Show that the polar form of AB in Example A4 is equivalent to its rectangular
form
A7 Show that the polar form of A B in Example A4 is equivalent to its rectangular
form
A8 Using Euler s formula, show that sin = (ej e j )/2j
A P P E N D I X
Fundamentals of Engineering (FE) Examination
B1 INTRODUCTION
The Fundamentals of Engineering (FE) examination1 is one of four steps to be completed toward registering as a Professional Engineer (PE) Each of the 50 states in the United States has laws that regulate the practice of engineering; these laws are designed to ensure that registered professional engineers have demonstrated suf cient competence and experience The same exam is administered at designated times throughout the country, but each state s Board of Registration administers the exam and supplies information and registration forms You will nd a Web reference to the phone numbers of the Boards of Registration in the 50 states in the accompanying CD-ROM The four steps required to become a Professional Engineer are: 1 Education Usually satis ed by completing a BS degree in engineering from an accredited college or university 2 Fundamentals of Engineering Examination An 8-hour examination described in the next section of this appendix 3 Experience Following successful completion of the Fundamentals of Engineering Examination, 2 to 4 years of engineering experience are required
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