MA100 Lecture Note PDF

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Summary

Contents Preface 1 Orientation 1 Teaching materials and arrangements 1 Current exam structure for MA100 1 Maple 1 Mathematical background 1 Syllabus 2 The Logical Framework 2 Definition, proposition, proof and related terminology. 2 Truth tables, negations and compound propositions 2 Logical Equival...

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Contents Preface

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1 Orientation

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1.1 Teaching materials and arrangements . . . . . . . . . . . . . . . . . . . . . 1.2 Current exam structure for MA100 . . . . . . . . . . . . . . . . . . . . . .

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1.3 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.5 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 The Logical Framework 2.1 Definition, proposition, proof and related terminology . . . . . . . . . . . .

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2.2 Truth tables, negations and compound propositions . . . . . . . . . . . . . 2.3 Logical Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Predicates and quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 16 18

2.5 Some methods of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . .

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3 One-Variable Calculus, Part 1 of 7 3.1 Sets, subsets and intervals . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2 3.3 3.4 3.5 3.6

Functions, domain, codomain and range . . The set R2 , Cartesian equations and graphs Surjective, injective and bijective functions . The derivative . . . . . . . . . . . . . . . . . Continuity and differentiability . . . . . . .

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3.7 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . .

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4 One-Variable Calculus, Part 2 of 7

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4.1 Derivatives of elementary functions . . . . . . . . . . . . . . . . . . . . . . 4.2 Sum, product, quotient and chain rules . . . . . . . . . . . . . . . . . . . .

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4.3 Higher order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Taylor polynomials and Taylor series . . . . . . . . . . . . . . . . . . . . .

40 41

4.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . .

44 45

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5 One-Variable Calculus, Part 3 of 7 5.1 Increasing and decreasing functions . . . . . . . . . . . . . . . . . . . . . .

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5.2 Local extrema and strict local extrema . . . . . . . . . . . . . . . . . . . . 5.3 Stationary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4 Tests for classifying stationary points . . . . . . . . . . . . . . . . . . . . . 5.5 Convex and concave functions . . . . . . . . . . . . . . . . . . . . . . . . .

50 53

5.6 Inf lection points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.7 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.8 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 57 6 One-Variable Calculus, Part 4 of 7 6.1 Asymptotes and graph sketching . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Conics in standard position and orientation . . . . . . . . . . . . . . . . .

57 57 62

6.3 Global extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 68

6.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . .

72 72

7 One-Variable Calculus, Part 5 of 7

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7.1 The inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Some elementary bijective functions and their inverses . . . . . . . . . . . .

72 74

7.3 The derivative of an inverse function . . . . . . . . . . . . . . . . . . . . . 7.4 The local inverse of a non-bijective function . . . . . . . . . . . . . . . . .

81 83

7.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . .

85 86

8 One-Variable Calculus, Part 6 of 7

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8.1 Indefinite and definite integrals . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . 8.3 Primitives of elementary functions . . . . . . . . . . . . . . . . . . . . . . .

86 87 88

8.4 Integration by recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Integration by change of variable . . . . . . . . . . . . . . . . . . . . . . .

89 92

8.6 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 96

8.8 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . .

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9 One-Variable Calculus, Part 7 of 7 9.1 Integration by partial fractions . . . . . . . . . . . . . . . . . . . . . . . . .

98 98

9.2 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.3 Consumers’ and producers’ surplus . . . . . . . . . . . . . . . . . . . . . . 105 9.4 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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9.5 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 107 10 Matrices, Part 1 of 3

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10.1 Definitions, notation and terminology . . . . . . . . . . . . . . . . . . . . . 107 10.2 Operations on matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 10.3 The laws of matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10.4 The inverse matrix and its properties . . . . . . . . . . . . . . . . . . . . . 114 10.5 Powers of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 10.6 Properties of the transpose of a matrix . . . . . . . . . . . . . . . . . . . . 115 10.7 Matrix equations and their solutions . . . . . . . . . . . . . . . . . . . . . 116 10.8 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.9 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 117 11 Matrices, 2 of 3

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11.1 Solving systems of n linear equations for n unknowns . . . . . . . . . . . . 118 11.2 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 11.3 Elementary matrices and row-equivalence . . . . . . . . . . . . . . . . . . . 123 11.4 Theorems on matrix invertibility . . . . . . . . . . . . . . . . . . . . . . . 126 11.5 Inversion algorithm based on elementary row operations . . . . . . . . . . . 128 11.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 11.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 131 12 Matrices, 3 of 3

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12.1 Minors, cofactors and the determinant . . . . . . . . . . . . . . . . . . . . 131 12.2 The properties of the determinant . . . . . . . . . . . . . . . . . . . . . . . 135 12.3 Calculating determinants using row operations . . . . . . . . . . . . . . . . 138 12.4 Inverting a matrix using cofactors and the determinant . . . . . . . . . . . 140 12.5 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 12.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 12.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 144 13 Developing Geometric Insight, 1 of 2 13.1 Visualising the set R2 using position vectors . . . . . 13.2 Visualising vector operations in R2 . . . . . . . . . . 13.3 Lines in R2 . . . . . . . . . . . . . . . . . . . . . . . 13.4 Geometric and algebraic approaches to linear systems

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144 144 145 148 155

13.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 13.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 157

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14 Developing Geometric Insight, 2 of 2 158 3 14.1 Visualising vectors and operations in R . . . . . . . . . . . . . . . . . . . 158 14.2 Planes in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 14.3 Lines in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 14.4 Geometric and algebraic approaches to linear systems . . . . . . . . . . . . 169 14.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 14.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 175 15 Systems of Linear Equations, 1 of 2 175 15.1 Systems of m linear equations for n unknowns . . . . . . . . . . . . . . . . 175 15.2 The Gauss-Jordan method . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 15.3 Solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 15.4 Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 15.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 15.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 181 16 Systems of Linear Equations, 2 of 2

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16.1 The principle of linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 16.2 The rank of a matrix and the main theorem revisited . . . . . . . . . . . . 184 16.3 Analysing the set of solutions of Ax = b . . . . . . . . . . . . . . . . . . . 185 16.4 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 16.5 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 188 17 Vector Spaces, 1 of 4 189 17.1 Definition of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 17.2 Subspaces and the subspace criterion . . . . . . . . . . . . . . . . . . . . . 191 17.3 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 17.4 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 194 18 Vector Spaces, 2 of 4 195 18.1 Linear span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 18.2 Linear independence and linear dependence . . . . . . . . . . . . . . . . . 198 18.3 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 18.4 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 202 19 Vector Spaces, 3 of 4 202 19.1 Theorems on linear span and linear independence . . . . . . . . . . . . . . 202 19.2 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 19.3 Coordinates with respect to a basis . . . . . . . . . . . . . . . . . . . . . . 207 19.4 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 19.5 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 210

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20 Vector Spaces, 4 of 4 210 20.1 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 20.2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 20.3 The Cauchy-Schwarz inequality . . . . . . . . . . . . . . . . . . . . . . . . 213 20.4 Angle and orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 20.5 Pythagoras’ theorem and the triangle inequality . . . . . . . . . . . . . . . 214 20.6 Orthonormality and the Gram-Schmidt process . . . . . . . . . . . . . . . 216 20.7 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 20.8 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 218

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Preface These lecture notes are intended as a self-contained study resource for the MA100 Mathematical Methods course at the LSE. At the same time, they are designed to complement the MA100 course texts, ‘Linear Algebra, Concepts and Methods’ by Martin Anthony and Michele Harvey, and ‘Calculus, Concepts and Methods’ by Ken Binmore and Joan Davies. I am grateful to Martin Anthony and Michele Harvey for allowing me to use some materials from their ‘Linear Algebra, Concepts and Methods’ textbook and to Michele Harvey for commenting on a draft of the Calculus part of these lecture notes. I am also grateful to Siri Kouletsis for her invaluable help with typing and editing the manuscript and for various improvements to its content.

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1 1.1

Orientation Teaching materials and arrangements

Lecture notes The lecture notes are intended as your primary source of reading. They are self-contained and very detailed and include self-study exercises whose solutions can be found on Moodle. There is a set of lecture notes for the MT (Michaelmas Term) and another such set for the LT (Lent Term). The first set consists of sections 1 to 20 and covers weeks 1 to 10 of the MT. The second set consists of sections 21 to 39 and covers weeks 1 to 10 of the LT. The course proceeds at a constant pace, covering two sections per week. The current section, section 1, is an orientation section. It provides information about the teaching materials and arrangements and explains the role of each of the course components: the prerecorded lecture videos, the video slides, the live online lecture, the live online extra example sessions, the course forum, the classes, the class forums, the office hours, and the Mathematics Support Centre. Sections 2 to 39 cover the course syllabus and section 40 is a revision section. Prerecorded lecture videos and video slides Each section of the lecture notes is supported by prerecorded videos that follow the ordering of the subsections in the section. You can find the videos pertaining to a section and the corresponding video slides on Moodle under the relevant week. The role of each video varies depending on the section and subsection to which it belongs. Sometimes a video may provide additional theory to facilitate visualisation and understanding of a key concept. Other times it may provide a brief introduction to a subsection or a summary of important ideas, or illustrate the use of a theorem or the application of a method. In my opinion, a good approach is to read a section of the lecture notes before you watch the videos (so that you get familiar with main ideas and definitions) and then study this section thoroughly after you have watched the videos. If you understand a section from your first reading, you may skip the videos; however, you should then carefully go through the video slides in order to check that you have not missed anything important. If you decide at that point to watch a part of a video, you can quickly locate that part using the slides. Live online lecture Every week, there is a live online lecture taking place on Tuesday at 14:00. The aim of this session is to help you consolidate your understanding of the lecture notes and lecture videos by elaborating on the most challenging parts of the week’s material. In addition, I will use part of this session to address any questions you have posted on the course forum. The relevant Zoom link is a recurring one and can be found on MA100 Moodle.

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Live online extra examples sessions There are two online extra examples sessions every week, taking place on Thursday at 11:00 and at 15:00. The role of these sessions is to answer any additional questions you have posted on the course forum (that have not been answered by Tuesday) and also provide further examples and exercises. Typically, I will give you a few minutes to solve a part of an exercise and then I will solve that part live, answering any related follow-up questions. The recurring Zoom links for these sessions can also be found on MA100 Moodle. The course forums There are two course forums that can be accessed via MA100 Moodle. One of the forums is for general announcements I will be making throughout the year. The other forum is intended for any questions you may have about the course (as mentioned above), and your fellow students can contribute to the discussions. The classes These are weekly sessions where you strengthen your understanding of the course material with the help of your class teacher. Your first class in each term takes place in week 2 and your last class in week 11. Some of these classes take place on campus and some are conducted online, but they all have the same structure and cover the same material. Each student is assigned to a class group of about fifteen students who usually follow the same degree programme. Your personal timetable indicates the day and time of your class along with the room number (if the class takes place on campus). Please note that class attendance is compulsory and is recorded by your class teacher. The class forums There is a forum dedicated to each class group where you can post questions about the selfstudy exercises and related theory (in preparation for your class) as well as clarifications about the class exercises and other class materials (after your class). Your classmates can contribute to the discussions and your class teacher will monitor the forum and intervene as needed. In-class practice exercise Each class contains an open-book in-class practice session, taking place in the last fifteen minutes of each class. Your class teacher will give you an ‘in-class practice exercise’ to solve and you can use the lecture notes and your own notes as needed, but you have to try to work individually. Your class teacher will provide general hints as to how to approach the given exercise and also provide individual help if needed. The idea behind this exercise is to give you the opportunity to evaluate on a weekly basis where you stand in relation to the demands and pace of the course. Your solution should be submitted electronically to your teacher via MA100 Moodle by the end of the day on which your class takes place. Information regarding how to submit 8

your in-class practice exercise is provided on Moodle. Your class teacher will receive your work electronically and give you collective feedback on your work via your class forum, emphasising key ideas and drawing your attention to common mistakes and misconceptions. Class structure A typical class consists of three parts: In the first part of the class, your teacher will give you a ‘consolidation exercise’ in order to ensure that the main topics covered in the previous week’s class have been understood. Your teacher will provide the solution to this exercise afterwards. In the second part of the class, the focus shifts to the most recent material, that is, the material corresponding to the previous week’s lecture notes and videos. Your teacher will typically answer questions you may have about the theory, or expand on a ...