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Unformatted text preview: Section 5.3 Complex Vector Spaces & Appendix B Definition: A complex number is an ordered pair of real numbers, denoted either by ( a,b ) or by a + bi where i = √- 1 . Usual notation is z = a + bi , a=Re(z) is called the real part of z, b=Im(z) is called the imaginary part of z. Set of complex numbers is denoted by C . Definition: Two complex numbers. a + bi and c + di are defined to be equal written as a + bi = c + di if a=c and b=d. Addition, Substraction, Multiplication and Division • Addition: (a+bi)+(c+di)=a+c+(b+d)i • Substraction: (a+bi)-(c+di)=a-c+(b-d)i • Scalar multiplication: k(a+bi)=ka+kbi, k is a real number • Multiplication: (a+bi)(c+di)=(ac-bd)+(ad+bd)i, ( i 2 =- 1 ) Example: Given z 1 = 2- 5 i and z 2 =- 1- i . Find (a) iz 1- z 1 z 2 (b) ( z 1 + 1 + 3 z 2 ) 2 Solution: (a) i(2-5i)-(2-5i)(-1-i)=(2-5i)(i-(-1-i))=(2-5i)(2i+1)=12-i (b) [(2- 5 i ) + 1 + 3(- 1- i )] 2 = [3- 5 i- 3- 3 i ] 2 = (- 8 i ) 2 =- 64 To define division we need Definition: If z=a+bi is any complex number, then the complex conjugate of z is...
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This document was uploaded on 04/26/2011.
- Spring '11