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# Linear Algebra As an Introduction to Abstract Mathematics Lecture Notes for MAT67 University of California, Davis written Fall 2007, last updated

Only one question help needed. Please. Chapter 11, Proof writing Exercises q1.

Linear Algebra As an Introduction to Abstract Mathematics Lecture Notes for MAT67 University of California, Davis written Fall 2007, last updated February 10, 2012 Isaiah Lankham Bruno Nachtergaele Anne Schilling Copyright c c 2007 by the authors. These lecture notes may be reproduced in their entirety for non-commercial purposes.
Contents 1 What is Linear Algebra? 1 1.1 Introduction to MAT 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 What is Linear Algebra? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Non-linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.3 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.4 Applications of linear equations . . . . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Introduction to Complex Numbers 11 2.1 DeFnition of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations on complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Addition and subtraction of complex numbers . . . . . . . . . . . . . 12 2.2.2 Multiplication and division of complex numbers . . . . . . . . . . . . 13 2.2.3 Complex conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.4 The modulus (a.k.a. norm, length, or magnitude) . . . . . . . . . . . 16 2.2.5 Complex numbers as vectors in R 2 . . . . . . . . . . . . . . . . . . . 18 2.3 Polar form and geometric interpretation for C . . . . . . . . . . . . . . . . . 19 2.3.1 Polar form for complex numbers . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Geometric multiplication for complex numbers . . . . . . . . . . . . . 20 2.3.3 Exponentiation and root extraction . . . . . . . . . . . . . . . . . . . 21 2.3.4 Some complex elementary functions . . . . . . . . . . . . . . . . . . . 22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ii
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