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MATH1251 Mathematics for Actuarial Studies and Finance 1B ALGEBRA PROBLEMS Semester 2 2015 Copyright 2015 School of Mathematics and Statistics, UNSW
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Contents 6 COMPLEX NUMBERS 1 6.1 A review of number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.2 Introduction to complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.3 The rules of arithmetic for complex numbers . . . . . . . . . . . . . . . . . . . . . . 1 6.4 Real parts, imaginary parts and complex conjugates . . . . . . . . . . . . . . . . . . 1 6.5 The Argand diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.6 Polar form, modulus and argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.7 Properties and applications of the polar form . . . . . . . . . . . . . . . . . . . . . . 1 6.8 Trigonometric applications of complex numbers . . . . . . . . . . . . . . . . . . . . . 1 6.9 Geometric applications of complex numbers . . . . . . . . . . . . . . . . . . . . . . . 1 6.10 Complex polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.11 Stability of discrete and continuous time systems . . . . . . . . . . . . . . . . . . . . 1 6.12 Appendix: A note on proof by induction . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.13 Appendix: The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.14 Complex numbers and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 VECTOR SPACES 15 7.1 Definitions and examples of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 15 7.2 Vector arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.3 Subspaces of R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.4 Linear combinations and spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.5 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.6 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.7 Coordinate vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.8 Further important examples of vector spaces . . . . . . . . . . . . . . . . . . . . . . 15 7.9 Data fitting and polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . 15 7.10 Appendix: A brief review of set and function notation . . . . . . . . . . . . . . . . . 15 7.11 Vector spaces and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8 LINEAR TRANSFORMATIONS 33 8.1 Introduction to linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.2 Linear maps from R n to R m and m × n matrices . . . . . . . . . . . . . . . . . . . . 33 iii
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8.3 Geometric examples of linear transformations . . . . . . . . . . . . . . . . . . . . . . 33 8.4 Subspaces associated with linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.5 Further applications and examples of linear maps . . . . . . . . . . . . . . . . . . . . 33 8.6 Representation of linear maps by matrices . . . . . . . . . . . . . . . . . . . . . . . . 33 8.7 Matrix arithmetic and linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.8 One-to-one, onto and invertible linear maps and matrices . . . . . . . . . . . . . . . 33 8.9 Proof of the Rank-Nullity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.10 One-to-one, onto and inverses for functions . . . . . . . . . . . . . . . . . . . . . . . 33 8.11 Linear transformations and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 9 EIGENVALUES AND EIGENVECTORS 47 9.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 9.2 Eigenvectors, bases, and diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . 47 9.3 Eigenvalues of matrices with special structure . . . . . . . . . . . . . . . . . . . . . . 47 9.4 Applications of eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 47 9.5 Markov systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 9.6 Eigenvalues and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ANSWERS TO SELECTED PROBLEMS 55 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 PAST CLASS TESTS 79 iv
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Chapter 6 COMPLEX NUMBERS 6.1 A review of number systems 6.2 Introduction to complex numbers 6.3 The rules of arithmetic for complex numbers 6.4 Real parts, imaginary parts and complex conjugates 6.5 The Argand diagram 6.6 Polar form, modulus and argument 6.7 Properties and applications of the polar form 6.8 Trigonometric applications of complex numbers 6.9 Geometric applications of complex numbers 6.10 Complex polynomials 6.11 Stability of discrete and continuous time systems 6.12 Appendix: A note on proof by induction 6.13 Appendix: The Binomial Theorem 6.14 Complex numbers and Matlab 1
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2 CHAPTER 6. COMPLEX NUMBERS Problems for Chapter 6 “Why,” said the Dodo, “the best way to explain it is to do it.” - Lewis Carroll, Alice in Wonderland.
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