MATH1131 1141 algebra notes PDF

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MATH1131 Mathematics 1A and

MATH1141 Higher Mathematics 1A ALGEBRA NOTES CRICOS Provider No: 00098G

 c 2018 School of Mathematics and Statistics, UNSW Sydney

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Preface Please read carefully. These Notes form the basis for the algebra strand of MATH1131 and MATH1141. However, not all of the material in these Notes is included in the MATH1131 or MATH1141 algebra syllabuses. A detailed syllabus is given, commencing on page (ix) of these Notes. In using these Notes, you should remember the following points: 1. Most courses at university present new material at a faster pace than you will have been accustomed to in high school, so it is essential that you start working right from the beginning of the session and continue to work steadily throughout the session. Make every effort to keep up with the lectures and to do problems relevant to the current lectures. 2. These Notes are not intended to be a substitute for attending lectures or tutorials. The lectures will expand on the material in the notes and help you to understand it. 3. These Notes may seem to contain a lot of material but not all of this material is equally important. One aim of the lectures will be to give you a clearer idea of the relative importance of the topics covered in the Notes. 4. Use the tutorials for the purpose for which they are intended, that is, to ask questions about both the theory and the problems being covered in the current lectures. 5. Some of the material in these Notes is more difficult than the rest. This extra material is marked with the symbol [H]. Material marked with an [X] is intended for students in MATH1141. 6. Problems marked with [V]have a video solution available from Moodle. 7. It is essential for you to do problems which are given at the end of each chapter. If you find that you do not have time to attempt all of the problems, you should at least attempt a representative selection of them. The problems set in tests and exams will be similar to the problems given in these notes. Further information on the problems and class tests is on pages (x) and (240). 8. You will be expected to use the computer algebra package Maple in tests and understand Maple syntax and output for the end of semester examination. Note. 2018 School of Mathematics and Statistics, UNSW Sydney c

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CHAPTER 0. PREFACE

These notes have been prepared by many members of the School of Mathematics and Statistics. The main contributors include Peter Blennerhassett, Peter Brown, Shaun Disney, Peter Donovan, Ian Doust, David Hunt, Chi Mak, Elvin Moore and Colin Sutherland. Copyright is vested in The c University of New South Wales, 2018.

2018 School of Mathematics and Statistics, UNSW Sydney c

CONTENTS

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Contents Preface iii Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Syllabus for MATH1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Syllabus for MATH1141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Homework schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Tutorial schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 INTRODUCTION TO VECTORS 1.1 Vector quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Geometric vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Two dimensional vector quantities . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vector quantities and Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Vectors in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Rn and analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 n-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Lines in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Lines in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Lines through two given points (in Rn ) . . . . . . . . . . . . . . . . . . . . . . 1.5 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Linear combination and span . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Parametric vector form of a plane . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Cartesian form of a plane in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Vectors and Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Problems for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 VECTOR GEOMETRY 45 2.1 Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 The dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2.1 Arithmetic properties of the dot product . . . . . . . . . . . . . . . . . . . . . 48 2.2.2 Geometric interpretation of the dot product in Rn . . . . . . . . . . . . . . . 48 2.3 Applications: orthogonality and projection . . . . . . . . . . . . . . . . . . . . . . . . 50 2018 School of Mathematics and Statistics, UNSW Sydney c

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2.4

2.5 2.6

2.7

2.3.1 Orthogonality of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3.3 Distance between a point and a line in R3 . . . . . . . . . . . . . . . . . . . . 55 The cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.1 Arithmetic properties of the cross product . . . . . . . . . . . . . . . . . . . . 58 2.4.2 A geometric interpretation of the cross product . . . . . . . . . . . . . . . . . 59 2.4.3 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Scalar triple product and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.5.1 Volumes of parallelepipeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Planes in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6.1 Equations of planes in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6.2 Distance between a point and a plane in R3 . . . . . . . . . . . . . . . . . . . 69 Geometry and Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Problems for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3 COMPLEX NUMBERS 81 3.1 A review of number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2 Introduction to complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 The rules of arithmetic for complex numbers . . . . . . . . . . . . . . . . . . . . . . 84 3.4 Real parts, imaginary parts and complex conjugates . . . . . . . . . . . . . . . . . . 86 3.5 The Argand diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.6 Polar form, modulus and argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.7 Properties and applications of the polar form . . . . . . . . . . . . . . . . . . . . . . 94 3.7.1 The arithmetic of polar forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.7.2 Powers of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.7.3 Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.8 Trigonometric applications of complex numbers . . . . . . . . . . . . . . . . . . . . . 102 3.9 Geometric applications of complex numbers . . . . . . . . . . . . . . . . . . . . . . . 105 3.10 Complex polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.10.1 Roots and factors of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.10.2 Factorisation of polynomials with real coefficients . . . . . . . . . . . . . . . . 112 3.11 Appendix: A note on proof by induction . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.12 Appendix: The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.13 Complex numbers and Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4 LINEAR EQUATIONS AND MATRICES 129 4.1 Introduction to linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2 Systems of linear equations and matrix notation . . . . . . . . . . . . . . . . . . . . 133 4.3 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.1 Interchange of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.3.2 Adding a multiple of one equation to another . . . . . . . . . . . . . . . . . . 138 4.3.3 Multiplying an equation by a non-zero number . . . . . . . . . . . . . . . . . 139 4.4 Solving systems of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.4.1 Row-echelon form and reduced row-echelon form . . . . . . . . . . . . . . . . 140 2018 School of Mathematics and Statistics, UNSW Sydney c

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4.5 4.6 4.7 4.8

4.9

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4.4.2 Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.4.3 Transformation to reduced row-echelon form . . . . . . . . . . . . . . . . . . 145 4.4.4 Back-substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Deducing solubility from row-echelon form . . . . . . . . . . . . . . . . . . . . . . . . 149 Solving Ax = b for indeterminate b . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 General properties of the solution of Ax = b . . . . . . . . . . . . . . . . . . . . . . 151 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.8.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.8.2 Chemical engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.8.3 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Matrix reduction and Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5 MATRICES 173 5.1 Matrix arithmetic and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.1.1 Equality, addition and multiplication by a scalar . . . . . . . . . . . . . . . . 174 5.1.2 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.1.3 Matrix arithmetic and systems of linear equations . . . . . . . . . . . . . . . 182 5.2 The transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2.1 Some uses of transposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.2.2 Some properties of transposes . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.3 The inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.3.1 Some useful properties of inverses . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.3.2 Calculating the inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . 190 5.3.3 Inverse of a 2 × 2 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.3.5 Inverses and solution of Ax = b . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.4.1 The definition of a determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.4.2 Properties of determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.4.3 The efficient numerical evaluation of determinants . . . . . . . . . . . . . . . 201 5.4.4 Determinants and solutions of Ax = b . . . . . . . . . . . . . . . . . . . . . . 204 5.5 Matrices and Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 ANSWERS Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5

TO SELECTED PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ALGEBRA SYLLABUS AND LECTURE TIMETABLE The algebra course for both MATH1131 and MATH1141 is based on the MATH1131/MATH1141 Algebra Notes that are included in the Course Pack. Please note that the order of the syllabus changed in 2014, in accordance with requests from the Engineering Faculty and the School of Physics. It is important to note this in regard to the class tests from previous years. The computer package Maple will be used in the algebra course. An introduction to Maple is included in the booklet Computing Laboratories Information and First Year Maple Notes . The lecture timetable is given below. Lecturers will try to follow this timetable, but some variations may be unavoidable, especially in MATH1141 classes and lecture groups affected by public holidays. Chapter 1. Introduction to Vectors Lecture 1. Vector quantities and Rn . (Section 1.1, 1.2). Lecture 2. R2 and analytic geometry. (Section 1.3). Lecture 3. Points, line segments and lines. Parametric vector equations. Parallel lines.(Section 1.4). Lecture 4. Planes. Linear combinations and the span of two vectors. Planes though the origin. Parametric vector equations for planes in Rn . The linear equation form of a plane. (Section 1.5). Chapter 2. Vector Geometry

Lecture 5. Length, angles and dot product in R2 , R3 , Rn . (Sections 2.1,2.2). Lecture 6. Orthogonality and orthonormal basis, pro jection of one vector on another. Orthonormal basis vectors. Distance of a point to a line. (Section 2.3). Lecture 7. Cross product: definition and arithmetic properties, geometric interpretation of cross product as perpendicular vector and area (Section 2.4). Lecture 8. Scalar triple products, determinants and volumes (Section 2.5). Equations of planes in R3 : the parametric vector form, linear equation (Cartesian) form and point-normal form of equations, the geometric interpretations of the forms and conversions from one form to another. Distance of a point to a plane in R3 . (Section 2.6). Chapter 3. Complex Numbers

Lecture 9. Development of number systems and closure. Definition of complex numbers and of complex number addition, subtraction and multiplication. (Sections 3.1, 3.2, start Section 3.3). Lecture 10. Division, equality, real and imaginary parts, complex conjugates. (Finish 3.3, 3.4). Lecture 11. Argand diagram, polar form, modulus, argument. (Sections 3.5, 3.6). Lecture 12. De Moivre’s Theorem and Euler’s Formula. Arithmetic of polar forms. (Section 3.7, 3.7.1). Lecture 13. Powers and roots of complex numbers. Binomial theorem and Pascal’s triangle. (Sections 3.7.2, 3.7.3, start Section 3.8). Lecture 14. Trigonometry and geometry. (Finish 3.8, 3.9). Lecture 15. Complex polynomials. Fundamental theorem of algebra, factorization theorem, factorization of complex polynomials of form z n − z0 , real linear and quadratic factors of real polynomials. (Section 3.10).

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CONTENTS Chapter 4. Linear Equations and Matrices

Lecture 16. Introduction to systems of linear equations. Solution of 2 × 2 and 2 × 3 systems and geometrical interpretations. (Section 4.1). Lecture 17. Matrix notation. Elementary row operations. (Sections 4.2, 4.3). Lecture 18. Solving systems of equations via Gaussian elimination. (Section 4.4) Lecture 19. Deducing solubility from row-echelon form. Solving systems with indeterminate right hand side. (Section 4.5, 4.6). Lecture 20. General properties of solutions to Ax = b. (Section 4.7). Applications. (Section 4.8) or Matrix operations (start Section 5.1) Chapter 5. Matrices

Lecture 21. Operations on matrices. Transposes. (Sections 5.1, 5.2). Lecture 22. Inverses and definition of determinants. (Section 5.3 and start Section 5.4). Lecture 23. Properties of determinants. (Section 5.4). EXTRA ALGEBRA TOPICS FOR MATH1141 Extra topics for MATH1141 in semester 1 may be selected from the following: Introduction to Vectors. Use of vectors to prove geometric theorems; parametric vector equations for rays, line segments, parallelograms, triangles; elements of vector calculus. Vector Geometry. Use of vectors to prove geometric theorems, further applications of vectors to physics and engineering. Complex Numbers. Cardan’s formula for roots of cubics, applications of complex numbers to vibrating systems. Linear Equations. Elementary matrices and elementary row operations, applications of linear equations and matrices to electrical engineering (Kirchhoff’s Laws), economics (Leontief model). Matrices and Determinants. Rotations of Cartesian coordinate systems and orthogonal matrices, evaluation of special determinants and connections with areas.

ALGEBRA PROBLEM SETS The Algebra problems are located at the end of each chapter of the Algebra Notes booklet. They are also available from the course module on the UNSW Moodle server. The problems marked [R] form a basic set of problems which you should try first. Problems marked [H] are harder and can be left until you have done the problems marked [R] . You do need to make an attempt at the [H] problems because problems of this type will occur on tests and in the exam. If you have difficulty with the [H] problems, ask for help in your tutorial. Questions marked with a [V] have a video solution available from the course page for this subject on Moodle. The problems marked [X] are intended for students in MATH1141 – they relate to topics which are only covered in MATH1141. Extra problem sheets for MATH1141 may be issued in lectures. There are a number of questions marked [M], indicating that Maple is required in the solution of the problem.

2018 School of Mathematics and Statistics, UNSW Sydney c

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xi WEEKLY ALGEBRA SCHEDULES

Solving problems and writing mathematics clearly are two separate skills that need to be developed through practice. We recommend that you keep a workbook to practice writing solutions to mathematical problems. The following table gives the range of questions suitable for each week. In addition it suggests specific recommended problems to do before your classroo...