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Unformatted text preview: Linear Algebra As an Introduction to Abstract Mathematics Lecture Notes for MAT67 University of California, Davis written Fall 2007, last updated November 19, 2009 Isaiah Lankham Bruno Nachtergaele Anne Schilling Copyright c circlecopyrt 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.4 Applications of linear equations . . . . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Introduction to Complex Numbers 11 2.1 Definition 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 3 The Fundamental Theorem of Algebra and Factoring Polynomials 26 3.1 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . 26 3.2 Factoring polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Vector Spaces 36 4.1 Definition of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Elementary properties of vector spaces . . . . . . . . . . . . . . . . . . . . . 38 4.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Sums and direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46...
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This note was uploaded on 06/06/2010 for the course MATH 67 taught by Professor Schilling during the Spring '07 term at UC Davis.

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mat67_course_notes_2up - Linear Algebra As an Introduction...

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