Outline

THE LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE

MA100

MATHEMATICAL METHODS

STUDY PACK

General Information

Teaching Arrangements

Course Materials

Background

Outline of Lectures

2014/15

General Information

Lecturer: Ioannis Kouletsis

Office: COL 4.13

Class Teachers’ Office: COL 4.15

Mathematics Departmental Office: COL 4.01

Teaching Arrangements

Lectures: There are two lectures each week, one on Tuesday

at 2pm and one on Friday at 11am. Both lectures take place

in the Peacock Theatre. During these lectures, I write notes

which are projected on the theatre’s screen. These notes are

rather brief. Their aim is to introduce you to the material

covered in this module and familiarise you with its practical

uses.

An extended typed version of these notes are posted on the

MA100Moodle webpage, accessed via https://moodle.lse.ac.uk

using your personal login username and password. These are

the lecture notes. Together with the course texts, the lecture

2

notes are intended as your primary source of reading. They

contain several examples, explanations and remarks as well

as practice questions, homework questions to be submitted to

your class teacher, exam-style questions and detailed references

to the course texts.

The lecture notes are divided into forty sections. Each section

corresponds to a lecture and is posted on the MA100 Moodle

webpage a day before that lecture. An outline of the lecture

notes, organised by sections and subsections, begins on page

12 of this study pack. This outline provides a summary of the

entire content of the course and spans twenty weeks of lectures.

Lectures are captured on video. Each video recording is posted

on the MA100 Moodle webpage within a day of the correspond-

ing lecture taking place. The video recordings consist of the

notes projected on the theatre’s screen and the accompanying

verbal comments.

Given that each lecture is recorded and that the corresponding

lecture notes are made available a day in advance, you may

find it preferable not to copy notes during lectures in order

to concentrate on what is being presented. Or you may still

prefer to take partial or even complete notes while listening to

the presentation of the material, if this works best for you. It

is also possible to read and print the lecture notes before each

lecture and use the printout for additional note-taking during

3

the lecture.

The lectures are optional but I would strongly recommend that

you attend them because they aim to facilitate your introduc-

tion to new, and sometimes complex, concepts. Moreover,

attending lectures allows you to gain an accurate sense of the

pace of the course. The material is not excessively difficult but

its volume should not be underestimated. Each section of the

lecture notes is about fifteen pages long, and there are forty

such sections. Attending lectures keeps you in tune with the

course - this protects you, to some extent, from falling behind

its pace.

It is essential that we maintain a quiet environment during

lectures. The quality of the lectures and their recordings dete-

riorates rapidly when the level of noise increases in the Peacock

Theatre.

Extra-Examples Sessions: These are optional interactive

sessions designed to reinforce the material covered in the lec-

tures. There is no new theory presented in these sessions. The

focus is placed on clarifying the main concepts and solving the

exam-style questions found in the lecture notes.

There is one extra-examples session each week. It takes place

each Thursday at 11am in the Old Theatre. The first such

session starts in the second week of the term and deals with

4

the material covered in lectures 2 and 3. Lecture 1 is only an

orientation lecture.

As with the lectures, extra-examples sessions are captured on

video. Each video recording is posted on the MA100 Moodle

webpage within a day of the corresponding session taking place.

In addition, solutions to the exam-style questions covered in

each extra-examples session are posted on the MA100 Moodle

webpage at the same time as the video recording.

Classes: These are interactive sessions where you discuss the

course material with your class teacher and go through the

homework questions found in the lecture notes. Class atten-

dance is obligatory and is recorded by your class teacher.

There is a class each week, starting in the second week of the

term. The material covered in the two lectures of a given week

is discussed in the class of the subsequent week. Since lecture

1 is an orientation lecture, the first class of the term deals only

with the material covered in lecture 2.

Each student is assigned to a class group of about fifteen stu-

dents which usually follow the same degree programme. Your

personal timetable indicates the day and time of your class

along with the room number.

Each section of the lecture notes contains two homework ques-

tions that need to be submitted to your class teacher. This

5

means that there are four homework questions to be prepared

each week for the subsequent week’s class. The first class of

the term is no exception to this rule, because lecture 2 comes

with four homework questions.

For your first class meeting, please bring your homework to

your classroom. This should be neatly stapled, with your name

and group number written on the front page. Your homework

will be marked by your class teacher and returned to you at

the beginning of the following week’s class. Your class teacher

will organise the details of all subsequent homework submission

according to what works best with their teaching schedule.

I would recommend that you start working on the homework

questions as soon as the relevant lecture notes are posted on

the MA100 Moodle webpage. It is important that you work

through all the homework questions to the best of your abil-

ity. The lecture notes and the course texts provide sufficient

material for this purpose.

If you cannot complete a question, you are welcome to visit

my office hours or the office hours of any MA100 class teacher.

You can also get help from a fellow student. Solutions to

the homework questions are posted on the MA100 Moodle

webpage each Friday, after all the classes for that particular

week have taken place.

6

Some of the homework questions are designed to be challeng-

ing, so do not worry if you cannot solve them on your first

attempt. This is natural and part of your learning process.

However, after these questions have been covered in class and

their solutions have been posted on the MA100 Moodle web-

page, make sure that you have fully understood them; that is,

solve these questions yourself without looking at the solutions.

This is really important in order to gain the understanding

needed for success in the exam. This comment applies to the

practice questions and the exam-style questions as well.

The marking of your homework is intended only as feedback, so

the grades that you receive in class do not count towards your

degree. However, your attendance and participation in class

and your weekly submission of homework are considered by

the School to be important components of your performance

in your degree. A record of your class performance is kept

throughout your studies and, along with your exam results,

forms the basis for academic references.

Office Hours: For any questions about the course mate-

rial, you can visit my office hours or the office hours of any

MA100 class teacher. These are posted on the Mathematics

Department website:

http://www.lse.ac.uk/maths/Courses/Office Hours.aspx

7

Forums: There are two forums for this course that can be

accessed via the MA100 Moodle webpage. One of them is for

general announcements I may be making throughout the year.

The other forum is intended for any questions you may have

about the course, and your fellow students may participate in

the discussion.

Email Communication: For all other issues related to the

course, you can email me personally at [email protected]

Course Materials

Study Pack: This is the document you are holding now. Fur-

ther copies are available from the Mathematics Departmental

Office, COL 4.01. The Study Pack can also be downloaded

from the MA100 Moodle webpage.

Lecture Notes: As already explained, each section of these

notes is posted on the MA100 Moodle webpage a day before

the corresponding lecture. An outline of the lecture notes can

be found at the end of this study pack.

The Course Texts: You are expected to have a copy of

each of the following two books. The lecture notes contain

8

detailed references to them:

•Martin Anthony and Michele Harvey, Linear Algebra, Con-

cepts and Methods, Cambridge University Press,

• Ken Binmore and Joan Davies, Calculus, Concepts and

Methods, Cambridge University Press.

Reference Texts: The following books provide additional

reading and exercises and can be found in the library:

• H. Anton, Elementary Linear Algebra (or the Applications

Version of this book by Anton and Rorres),

• D. Lay, Linear Algebra and its Applications,

• S. L. Salas and E. Hille, Calculus, One and Several Vari-

ables,

• Schaum Outline Series: Mathematics for Economists; Lin-

ear Algebra; Advanced Calculus; Differential and Integral

Calculus.

Video Recordings of Lectures: Each recording is posted

on the MA100 Moodle webpage a day after the corresponding

lecture.

Video Recordings of Extra Examples Sessions: Each

recording is posted on the MA100 Moodle webpage a day after

9

the corresponding session.

Solutions to the Practice Questions: These refer to

the practice questions found in the lecture notes. As with the

lecture notes, each set of solutions is posted on the MA100

Moodle webpage a day before the corresponding lecture. The

idea is to have these solutions available when you read the

lecture notes, because sometimes you may clarify a part of the

theory by going through a practice question and its solution.

Ideally, I would recommend that you look at the solutions only

after attempting the questions.

Solutions to the Homework Questions Submitted

in Class: These are posted on the MA100 Moodle webpage

each Friday, after all classes for that week have taken place.

Solutions to the Exam-Style Questions: Each set of

solutions is posted on the MA100 Moodle webpage one day

after the corresponding extra-examples session.

Past Exams and Solutions to Past Exams: Past ex-

amination papers from the last three years can be found on

the MA100 Moodle webpage. Their solutions will be released

on that webpage after all lectures have finished, at the start of

the Easter break.

10

Past papers are useful for familiarising yourself with the style

of the exam, but I would advise against using past papers

for the purpose of deducing which topics are likely to appear

in the forthcoming exam. The key to success in the exam

is understanding the course material and solving the practice

questions, homework questions and exam-style questions.

Maple Tutorial: Maple is a computer program which is used

throughout the course for visualising graphs and solutions to

certain exercises.

Before the end of the fourth week of this term, you need to

complete the Maple tutorial found on the MA100 Moodle web-

page. This is part of your preparation for a Maple session in

week 5. You can use any of the computer facilities offered by

the School.

Your class during the fifth week of this term is held in one

of the computer rooms. The room number is listed on your

personal timetable. There, you will use Maple to work through

an exercise set that will be given out in week 4, in section 8 of

the lecture notes.

Later sections of the lecture notes will contain additional prac-

tice questions where Maple can be useful. Note that Maple is

not part of the examinable material.

11

Background

It is recommended that you look through the review booklets

An Algebra Refresher and A Calculus Refresher which can

be found at the following web address:

www.maths.lse.ac.uk/Refreshers/

An exercise set based on material with which you should al-

ready be familiar can be found on the MA100 Moodle webpage.

Solutions to this set and a Maple file containing all the relevant

graphs are available on that webpage.

In your spare time, work through these exercises and consult

their solutions. If you have difficulties with any part of this

material, you can look it up in the Refreshers using the above

web address. Alternatively, make a note of any background

you are missing and visit my office hours or the office hours of

any MA100 class teacher.

Outline of Lectures

1. Introduction

1.1 Teaching arrangements

1.2 Course materials

12

1.3 The MA100 Moodle webpage

1.4 Some advice

1.5 Additional support

2. The Logical Framework

2.1 Definition, proposition, proof and related terminology

2.2 Truth tables, negations and compound propositions

2.3 Logical equivalence

2.4 Predicates and quantifiers

2.5 Some methods of proof

2.6 Practice questions

2.7 Exercises for submission in class

2.8 Exam-style question

2.9 Relevant sections from the textbooks

3. One-Variable Calculus, 1 of 7

3.1 Sets, subsets and intervals

3.2 Functions, domain, codomain and range

3.3 The set R2, Cartesian equations and graphs

3.4 Surjective, injective and bijective functions

3.5 The derivative

3.6 Continuity and differentiability


13





4.1 Derivatives of elementary functions

4.2 Sum, product, quotient and chain rules

4.3 Higher order derivatives

4.4 Taylor polynomials and Taylor series






5.1 Increasing and decreasing functions

5.2 Local extrema and strict local extrema

5.3 Stationary points

5.4 Tests for classifying stationary points

5.5 Convex and concave functions

5.6 Inflection points



14




6.1 Asymptotes and graph sketching

6.2 Conics in adapted coordinates

6.3 Global extrema

6.4 Unconstrained and constrained optimisation






7.1 Invertibility of a function

7.2 Inverses of linear, affine and power functions

7.3 Inverses of exponential functions

7.4 Inverses of trigonometric functions

7.5 The derivative of an inverse function

7.6 The local inverse




15



8.1 Indefinite and definite integrals

8.2 The fundamental theorem of calculus

8.3 Primitives of elementary functions

8.4 Integration by recognition

8.5 Integration by change of variable

8.6 Trigonometric substitution






9.1 Integration by partial fractions

9.2 Integration by parts

9.3 Consumers’ surplus and producers’ surplus

9.4 Integrals of probability density functions





16

10. Matrices, 1 of 3

10.1 Definitions

10.2 Matrix operations

10.3 Row echelon and reduced row echelon forms

10.4 Matrix equations and their solutions






11.1 Elementary matrices

11.2 Row-equivalence

11.3 Matrix invertibility

11.4 Inverting a matrix using row operations






12.1 Minors, cofactors and the determinant

12.2 The properties of the determinant

17

12.3 Inverting a matrix using cofactors and the determinant

12.4 Cramer’s rule





13. Developing Geometric Insight, 1 of 2

13.1 Visualising vectors and vector operations in R2

13.2 Length, angle and the inner product

13.3 Vectors in R3 and in Rn





14. Developing Geometric Insight, 2 of 2

14.1 Points and lines in R2

14.2 Points, lines and planes in R3

14.3 Independent equations and free parameters

14.4 Parametric and Cartesian descriptions of flats in Rn



18



15. Systems of Linear Equations, 1 of 2

15.1 Coefficient matrix and augmented matrix

15.2 Gaussian elimination

15.3 Consistent and inconsistent systems

15.4 Leading and non-leading variables





16. Systems of Linear Equations, 2 of 2

16.1 Solution sets

16.2 Homogeneous systems

16.3 The associated homogeneous system

16.4 The principle of linearity





19

17. Vector Spaces, 1 of 4

17.1 The axioms of a vector space

17.2 Linear combinations of vectors

17.3 Subspaces and affine subsets

17.4 The subspace criterion






18.1 Linear span

18.2 Linear independence

18.3 Basis and dimension






19.1 Coordinates of a vector relative to a basis

19.2 Finite and infinite dimensional vector spaces

19.3 Vector spaces of functions and sequences

20






20.1 Inner-product spaces

20.2 The Cauchy-Schwarz inequality

20.3 The generalised Pythagoras theorem

20.4 The Gram-Schmidt orthonormalisation process





21. Linear Transformations, 1 of 5

21.1 Definition

21.2 Matrix representation of a linear transformation

21.3 Rotations and reflections on R2

21.4 Linear transformations on abstract vector spaces




21



22.1 Kernel and range

22.2 The rank-nullity theorem

22.3 Invertibility






23.1 Change of basis and transition matrix

23.2 Similarity

23.3 Introduction to diagonalisation






24.1 Eigenvalues and eigenvectors

24.2 Characteristic polynomial and eigenspaces

22

24.3 Algebraic and geometric multiplicity

24.4 Diagonalisation






25.1 Symmetric matrices and orthogonal diagonalisation

25.2 Classification of quadratic forms

25.3 Conic sections in general coordinates





26. Multivariate Calculus, 1 of 7

26.1 Functions of two variables and their graphs

26.2 Partial derivatives and their geometric meaning

26.3 The tangent plane

26.4 Linear, affine and homogeneous functions

26.5 Euler’s formula for homogeneous functions

26.6 Vertical and horizontal sections

23






27.1 The gradient: adopting a geometric approach

27.2 The gradient: using implicit differentiation

27.3 Directional derivatives and their geometric meaning

27.4 The rate of change of a function






28.1 Functions of three or more variables and their graphs

28.2 The tangent hyperplane

28.3 Gradient, derivative and directional derivatives





24


29.1 Stationary points and systems of non-linear equations

29.2 The Taylor polynomial of degree two

29.3 Classification of stationary points

29.4 The eigenvalue test and the principal-minors test






30.1 Vector-valued functions and their graphs

30.2 The chain rule for vector-valued functions

30.3 Adapting the chain rule






31.1 Convex sets in multi-dimensional Euclidean spaces

31.2 Unconstrained and constrained optimisation

31.3 Ad hoc methods of optimisation

25






32.1 Optimisation subject to a single equality constraint

32.2 Lagrange’s method

32.3 Regarding the form of the Lagrangian

32.4 Applicability of Lagrange’s method

32.5 Lagrange’s theorem and suggested methodology

32.6 The Lagrange multiplier





33. Differential and Difference Equations, 1 of 6

33.1 Discrete and continuous compound interest

33.2 Nominal and effective rates of interest

33.3 Depreciation, future value and discounting

33.4 Arithmetic and geometric sequences and series


26





34.1 Complex numbers and Cartesian form

34.2 Polar exponential form

34.3 De Moivre’s and Euler’s formulae

34.4 The fundamental theorem of algebra






35.1 General solution of a differential equation

35.2 Initial conditions

35.3 Separable differential equations

35.4 Exact differential equations

35.5 Linear first-order differential equations

35.6 Integrating factors



27




36.1 Homogeneous differential equations

36.2 Change of variables

36.3 Differential equations of the form P (D)y(x) = 0






37.1 Differential equations of the form P (D)y(x) = f (x)

37.2 Limits of functions

37.3 Long-term behaviour of solutions






38.1 Difference equations of the form P (E)yx = 0

28

38.2 Difference equations of the form P (E)yx = fx

38.3 Limits of sequences

38.4 Long-term behaviour of solutions





39. Further Applications, 1 of 2

39.1 Solving systems of differential equations

39.2 Solving systems of difference equations





40. Further Applications, 2 of 2

40.1 Markov chains

40.2 The cobweb model





29

Date post:	08-Dec-2015
Category:	Documents
Upload:	rahul-dhanji-manji
View:	7 times
Download:	2 times

Outline

Documents