mat1302-lecture18

# mat1302-lecture18 - MAT 1302 Mathematical Methods II...

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Unformatted text preview: MAT 1302 - Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2010 – Lecture 18 Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 1 / 30 Announcements Third Assignment Solutions now posted on course webpage. Second Midterm Exam This Friday. Covers material up to and including Lecture 16 (determinants and their properties). Bring your student ID. Know your DGD number, write in pen. Last time: Complex numbers Today: Eigenvectors/eigenvalues Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 2 / 30 Complex numbers – review Definition (complex number) A complex number is a number written in the form z = a + bi where a is the real part of z (Re z ), b is the imaginary part of z (Im z ), i is formal symbol satisfying i 2 =- 1. Notation/terminology Real number: a complex number with zero imaginary part (i.e. a + 0 i ). Imaginary number: a complex number with zero real part (i.e. 0+ bi ). The set of complex numbers is denoted C . Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 3 / 30 Arithmetic with complex numbers – review Examples: 1 (2- 5 i ) + (4 + 8 i ) = 6 + 3 i 2 (- 4 + i )- (2- 4 i ) =- 6 + 5 i 3 (- 2)(8- 7 i ) =- 16 + 14 i 4 (2- i )(- 3 + 2 i ) =- 6 + 4 i + 3 i- 2 i 2 =- 6 + 7 i + 2 =- 4 + 7 i Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 4 / 30 Complex conjugation – review Definition (complex conjugation) The conjugate of z = a + bi ( a , b ∈ R ) is ¯ z = a- bi (we negate the imaginary part). Examples 1- 5- 7 i =- 5 + 7 i 2- 3 + 3 4 i =- 3- 3 4 i Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 5 / 30 Absolute values – review If z = a + bi , then z ¯ z = ( a + bi )( a- bi ) = a 2- abi + abi- b 2 i 2 = a 2 + b 2 ≥ Definition (absolute value or modulus) If z ∈ C , then the absolute value (or modulus ) is | z | def = √ z ¯ z . So if z = a + bi , then | z | = p a 2 + b 2 . | z | is always a nonnegative number. Examples 1 If z = 2- i , then | z | = p 2 2 + (- 1) 2 = √ 4 + 1 = √ 5. 2 If z = 3 + 5 i , then | z | = √ 3 2 + 5 2 = √ 9 + 25 = √ 34. Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 6 / 30 Geometric interpretation Every complex number a + bi corresponds to a point ( a , b ) (or vector) in the plane R 2 . 6- imaginary axis real axis s z = a + bi b a s ¯ z = a- bi- b | z | | z | = √ a 2 + b 2 is the distance of z from the origin (Pythagorean Theorem). Alistair Savage (uOttawa) MAT 1302 - Mathematical Methods II Winter 2010 – Lecture 18 7 / 30 Multiplicative inverses and division – review If z 6 = 0, then | z | > 0 and z · ¯ z | z | 2 = z ¯ z | z | 2 = | z | 2 | z | 2 = 1 and so we write 1 z = z- 1 = ¯ z | z | 2 ....
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## This note was uploaded on 12/04/2011 for the course MAT 1302 taught by Professor Adjaoud during the Spring '11 term at University of Ottawa.

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mat1302-lecture18 - MAT 1302 Mathematical Methods II...

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