Autumn 2015
F17CC1 Algebra A
Lecturer Prof Mark V Lawson
Room CMS21
Ext 3210
Email m.v.lawson[at]hw.ac.uk
| Tutorials (begin week 2) all in
CMS01 The last tutorial will be on 26th November Tutorial helpers: Abla Azalekor and David Bolea |
|
| Times |
Who |
| 1.15
to 2.15 |
Actuarial
students surnames A to L inclusive. |
| 2.15
to 3.15 |
Actuarial students surnames M to Z inclusive. |
| 3.15
to 4.15 |
Maths
students (those taking Maths in context). |
| 4.15
to 5.15 |
Maths
students (those not taking Maths in context). All MSAS
students. |
| Students
not in the above categories (others) please
attend whichever tutorial that appears on your timetable. |
|
Recommended book
M. V. Lawson,
Algebra and Geometry: an Introduction to University
Mathematics, to be published by CRC Press in 2016.A free PDF file of this book will be made available via VISION. All exercises will be taken from this book. Solutions to exercises available via VISION.
Further reading and additional exercises
R. Hammack, Book of proof, VCU Mathematics Textbook Series, 2009. This book can be downloaded for free here.
This is an excellent reference for material dealing with sets and counting, in addition for more information about proofs in mathematics.J. Olive, Maths: a student's survival guide, second edition, CUP, 2006.
This book is primarily designed for science students but is, in fact, a very useful source for a lot of A-level/(Advanced) Highers standard mathematics that you may have forgotten from school or never encountered. In addition, it contains material that I will cover from scratch in this course in Chapters 1, 2, 3, 6, 7, 10, 11.S. Lipschutz and M. Lipson, Discrete mathematics, second edition and onwards, Schaum's Outlines, McGraw-Hill.
As with all the Schaum books, this is an excellent place to look for worked examples and further exercises. Chapters 1, 5 and 11 seem the most relevant to this course.
S. Lipschutz and M. Lipson, Linear Algebra, third edition and onwards, Schaum's Outlines, McGraw-Hill.
Chapters 1, 2, 3 and 8 are covered in this course.A. Hirst and D. Singerman, Basic algebra and geometry, Pearson, 2001.
This is a textbook with the same intentions as this course and would be useful as a source of different approaches and further examples. It covers much the same ground as my course, though oddly does not cover the solution of linear equations by Gaussian elimination.C. McGregor, J. Nimmo, W. Sothers, Fundamentals of university mathematics, 3rd Revised Edition, Woodhead Publishing Ltd, 2010.
This covers all the algebra I cover. There are some interesting exercises that would be a good test of understanding. This book is also a useful reference for calculus.
Lecture notes
|
|||||
| Introduction (4 lectures) |
I. Sets and counting (4 lectures) |
II. Number theory (4 lectures) |
III. Complex nos and polynomials (5 lectures) |
IV. Matrices (7 lectures) |
V. Vectors (3 lectures) |
| Read Sections 4.3 and 4.2. Then read for background Chapters 1 and 2. Watch Fermat's last theorem on the BBC iplayer. Lecture 1 Lecture 2 Lecture 3 Lecture 4 |
Read Sections 3.1, 3.9 and 4.4. Lecture 5 Lecture 6 Lecture 7 Lecture 8 |
Read Sections 5.1, 5.2 and 5.3 Lecture 9 Lecture 10 Lecture 11 Lecture 12 |
Complex
numbers
Read
sections
6.1, 6.2 and 6.3 Lecture 13
Lecture 14 Polynomials Read Sections 7.1, 7.2, 7.3, 7.4 and 7.5 Lecture 15 Lecture 16 Lecture 17 |
The material in this section
is standard and admits of little variation in
presentation. For lectures notes, read the sections of my book as indicated or use one of the other recommended books. Read sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6 (characteristic polynomials, eigenvalues and the Cayley-Hamilton theorem only) and 8.7. Lecture 18. Definition of matrices. Sum and difference of matrices, multiplication by a scalar, the transpose. See pp219--221 of book. Lecture 19. Definition of matrix multiplication. Outline of matrix algebra. See pp 222--238. You can omit proofs. Lecture 20. Quiz 1. Lecture 21. Definition of invertible matrix and inverse. Definition of determinant and its key property. Calculating the inverse using the adjugate matrix. See pp 247--267. There is obviously more material in the book than is covered in this course. Be guided by the lectures. Quiz 2 handed out. Lecture 22. Solutions to Quiz 2. Calculating inverse using adjugates (conclusion). The characteristic polynomial, eigenvalues, the Cayley-Hamilton (or vice-versa) theorem. See pp 267--270. This is all that I cover in this course but if you want to know why these are important ideas read Section 8.6 of the book. Lecture 23. Solving systems of linear equations using elementary row operations. See pp 238--247. This will be concluded in the next lecture. Lecture 24. Conclusion to solving linear equations. When to use the matrix method. Blankinship's algorithm. Solutions to Quiz 3. |
Read Sections 9.1, 9.2, 9.3 and 9.4 Lecture 25 Lecture 26 Lecture 27 |
| You can
omit exercises marked with a *. You can omit exercises that require proofs. Do Exercises 4.3. Do Exercises 4.2. When you have finished these, you can attempt Exercises 2.3. There are elements of revision in many of these exercises. |
Do Exercises 3.1, 3.9 (omit
Question 2) and 4.4. Don't forget, you can continue with the previous exercises. |
Do Exercises 5.1, 5.2 and 5.3. |
Complex
numbers
Do Exercises
6.1, 6.2 and 6.3 Polynomials
Do exercises 7.2, 7.4, 7.5 |
Do Exercises 8.1 (omit Q11) 8.2 (omit Q8 and Q9) 8.3 8.4 (in Q1 simply apply the given definition) 8.5 (Q1 only) 8.6 (Q1,Q2 and Q3 only. In Q3 just calculate the characteristic polynomials and the eigenvalues). 8.7 |
Do exercises 9.1 9.2 9.4 |
Quizzes
Quiz 1 and solutions
Quiz 2 and solutions
Quiz 3 and solutions
Quiz 4 and solutions
Quiz 5 and solutions
Quiz 2 and solutions
Quiz 3 and solutions
Quiz 4 and solutions
Quiz 5 and solutions
2015 exam paper and solutions
It is not the policy of
the Mathematics Department to make more than one
past paper available
Comments.
The exam paper is divided into two parts.
Question 1, worth 40%, contains simple calculations and was included because of the lack of tests this year.
The remaining questions are lengthier calculations.
The only questions that required the slightest amount of thought were Question 7 and part (a) of Question 1.
All the other questions tested basic methods.
The Quiz questions covered virtually all the exam questions.
All the questions were a subset of the learning outcomes.
The average mark was 60% with the highest being 97%.
51% of students obtained an A or B, 35% of students obtained a C or D, and 11% of students obtained an E or F (fail grades).
(Percentages are rounded).
The commonest errors were caused by poor arithmetic (in some cases, excruciatingly poor arithmetic) combined
with a lack of basic checks of solutions.
Useful links
NumberphileCentre 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 this 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.
15.XI.2015