Autumn 2019

Edinburgh

F17CC Introduction to University Mathematics

Lecturer Prof Mark V Lawson

Room CM G12

Ext 3210

Email m.v.lawson[at]hw.ac.uk

Lectures
A draft of my book has been made available on VISION

Errata page for book here

Background reading: Chapters 1 and 2.

1. Combinatorics. Sections 3.1, 3.2 (first page only) and 3.9. Main goal: understand and use set notation. 7 lectures.

2. Polynomials. Sections 4.2, 4.3 and 4.4; Chapter 6; Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.9. Main goal: understand the nature of the roots of a polynomial including complex ones. 10 lectures.

3. Matrices. Chapter 8 omitting Section 8.7 and only the characteristic polynomials and their eigenvalues from Section 8.6. Main goal: solve systems of linear equations using elementary row operations. 7 lectures.

4. Vectors. Sections 9.1, 9.2, 9.3, 9.4. Main goal: find the vector equations of lines and planes. 4 lectures

Proofs. Read Chapter 2 for the idea of proofs. The lectures will deal mainly with proof by induction: Section 3.8. 1 lecture.

Syllabus

Check out the syllabus for learning outcomes

Week/dates	Tuesday 11.20	Thursday 10.20	Friday 10.20	Homework
Week 1 16th to 20th September	1. Combinatorics Lecture 1: introduction to the course Introduction to course	Lecture 2	Lecture 3	Do Exercises 1. They only require school maths
Week 2 23rd to 27th September	Lecture 4	Lecture 5	Lecture 6	Do Exercises 2. Do Exercises 3. Do Quiz 1.
Week 3 30th September to 4th October	Lecture 7 REVISION OF SECTION 1 END OF SECTION 1	2. Polynomials Lecture 8	Lecture 9	Ensure that you can do all 10 questions of Lecture 7. Exercises 4.
Week 4 7th October to 11th October	Lecture 10	Lecture 11	TEST 1 DURING LECTURE Section 1 only Test lasts 15 minutes You can stay as long as you like after that to complete it	Exercises 5. Exercises 6.
Week 5 14th October to 18th October	Lecture 12	Lecture 13	Lecture 14	Exercises 7.
Week 6 21st October to 25th October	Lecture 15	Lecture 16	Lecture 17 REVISION OF SECTION 2 END OF SECTION 2	Complete all questions in Lecture 17.
Week 7 28th October to 1st November	3. Matrices Lecture 18	Lecture 19	Lecture 20	Exercises 8.
Week 8 4th November to 8th November	Lecture 21	Lecture 22	TEST 2 DURING LECTURE Section 2 only Test lasts 15 minutes You can stay as long as you like after that to complete it	Exercises 9.
Week 9 11th November to 15th November	Lecture 23	Lecture 24 Revision questions at end of lecture notes END OF SECTION 3	4. Vectors Lecture 25 Geometric introduction to free vectors: vector addition, multiplication by a scalar, inner products and vector products.	Exercises 10.
Week 10 18th November to 22nd November	Lecture 26	Lecture 27	Lecture 28 END OF SECTION 4	Exercises 11. Exercises 12.
Week 11 25th November to 29th November	Lecture 29 Proof by induction	LECTURES ARE CONCLUDED

Assessment

Tests 20%

These are closed book tests for feedback.

Test 1. Week 4. Section 1. Test paper and solutions.

Test 2. Week 8. Section 2. Test paper and solutions.

Final Exam 80%

The goal of this course is to develop an understanding of the ideas and methods of university mathematics that will form the foundation for your further studies.

I am, however, required by my department to provide you with one past exam paper simply to show you the nature of the exam:

2019 Exam paper and solutions

Highest mark = 100; average mark = 65 ; lowest mark = 4.

THE LEARNING OUTCOMES IN THE SYLLABUS ARE EFFECTIVELY A LIST OF ALL POSSIBLE EXAM QUESTIONS. So, check out the syllabus above now.

The actual exam will consist of questions which are of the same type as the tutorial questions as well as proofs from the lectures, exercise sheets and tests.
I will also expect you to know definitions and the statements of theorems.

All students are individually responsible for finding out when and where their exams are.

Calculators, dictionaries etc in exams.

Exam Advice

1. READ each question thoroughly to ensure that you answer all parts of the question and that you have interpreted the question correctly.

2. COMMUNICATE your answers. Your solutions must be CLEAR and SELF-EXPLANATORY not written in an ideolect that only you know or adorned with hieroglyphics that only you can interpret. It is NOT the job of the marker to figure out what you meant or to detect the correct solutions amongst a sea of calculations. COMMUNICATION is not talking to yourself but expressing yourself to others.

3. CHECK your answers. This means checking that your solution answers the question originally posed. Do NOT ASSUME that your calculations are correct; do NOT ASSUME you have not made any mistakes. Your default position should be --- I might have made an error. If you find an error CORRECT IT.

Tutorials are on Thursday afternoons
Teaching assistants: Christina Lazaridou, Nikoletta Louca, Kieran Quaine

YOU WILL HAVE 1 TUTORIAL A WEEK in F17CC

Tutorials start in week 2

If your programme is not mentioned explicitly below
please go to the tutorial given in your timetable

Maths students means all students on a Maths or a Maths and/with degree NOT AMS students

PLEASE ATTEND THE CORRECT SESSION:
email me if there are work conflicts

Time	Who
13.20-14.10	AMS students: surnames A to J
14.20-15.10	AMS students: surnames K to Z
15.20-16.10	Maths Students: surnames A to K (except Maths with/and CS) All Maths with Finance students All MSAS students
16.20-17.10	Maths Students: surnames L to Z In addition, all Maths with/and CS

Remember: the final exam paper will be closely based on questions selected from the exercise sheets below

ALL QUESTIONS MUST BE ATTEMPTED WITHOUT LOOKING AT THE SOLUTIONS --- NOT EVEN FOR HINTS

PLEASE DOWNLOAD THESE QUESTIONS TO YOUR LAPTOP OR PHONE.
HARD COPIES WILL NOT BE PROVIDED.

Exercises 1
Warm up
exercises

Exercises 2
Section 1
Basic set theory

Exercises 3
Section 1
Basic counting

QUIZ 1

Exercises 4
Section 2
Basic algebra

Exercises 5
Section 2
The Binomial theorem

Exercises 6
Section 2
Complex numbers

Exercises 7
Section 2
Polynomials

Exercises 8
Section 3
Matrices

Exercises 9
Section 3
Matrices

Exercises 10
Section 4
Vectors

Exercises 11
Section 4
Vectors

QUIZ 2

Thought
Provoking
Questions

Question 4 requires knowledge of greatest common divisors . See Section 5.2 (not part of course).
Question 9 is a combination of material on page 326 and Ex 10.1 (not part of course).
If you are interested in mathematics both questions are worth studying.
Otherwise omit.

Solutions 1

Solutions 2

Solutions 3

Solutions

Solutions 4

Solutions 5

Solutions 6

Solutions 7

Solutions 8

Solutions 9

Solutions 10

Solutions 11

You can find the solutions
to these questions
in the solutions to
the book exercises on VISION

Further reading and additional exercises

R. Hammack, Book of proof, VCU Mathematics Textbook Series, 2009. This book can be downloaded for free here.

J. Olive, Maths: a student's survival guide, second edition, CUP, 2006.

M. V. Lawson, Algebra & Geometry: An introduction to university mathematics, CRC Press, 2016.

A PDF of a prepublication version of this book will be made freely available via VISION to all students registered for this course.
The book covers much more than the lecture course but I have made clear above the sections you need. The errata page for this book
can be found here.

S. Lipschutz and M. Lipson, Discrete mathematics, revised third edition and onwards, Schaum's Outlines, McGraw-Hill Education, 2009.

S. Lipschutz and M. Lipson, Linear Algebra, fifth edition, Schaum's Outlines, McGraw-Hill Education, 2013.

C. McGregor, J. Nimmo, W. Sothers, Fundamentals of university mathematics, 3rd Revised Edition, Woodhead Publishing Ltd, 2010.

STEP QUESTIONS

If you want to really develop your mathematical skills you need to attempt much more thought-provoking questions than I provide in this course. I recommend the following website

https://maths.org/step/welcome

and, in particular, the Advanced Problems in Mathematics book. A free PDF is available here where you can also buy the book itself. Be warned that the questions cover much wider ground than is covered
in this course. In addition, I would recommend the books of A. Gardiner.

Useful links

Numberphile

Centre for Innovation in Mathematics Teaching
University of Plymouth

Online maths calculator
This is a nice multi-purpose site by Milos Petrovic.
You can check many of the calculations covered in my course via this site.

Online matrix calculator

Another online matrix calculator

An online version of Book 1 of Euclid's Elements

The MacTutor history of mathematics

A history of calculating

Understanding math(s)
Excellent advice on learning and understanding mathematics by Peter Alfeld at the University of Utah.