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iii MAT1503CONTENTS
Page
PREFACE v
Introduction v
Prescribed Book vii
Overview of the Module vii
Study Guide ix
Learning Strategies for Mathematics; and in particular for this Module x
Assessment xii
CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS AND MATRICES 1
1.1 Introduction to Systems of Linear Equations 2
1.2 Gaussian Elimination 6
1.3 Matrices and Matrix Operations 19
1.4 Inverses; Rules of Matrix Arithmetic 22
1.5 Elementary Matrices and a Method for Finding the Inverse of A 24
1.6 Further Results on Systems of Equations and Invertibility 28
1.7 Diagonal, Triangular, and Symmetric Matrices 30
Review of Chapter 1 31
CHAPTER 2 DETERMINANTS 33
2.1 Determinants by Cofactor Expansion 34
2.2 Evaluating Determinants by Row Reduction 36
2.3 Properties of the Determinant Function 39
Review of Chapter 2 43
CHAPTER 3 VECTORS IN 2-SPACE AND 3-SPACE 45
3.1 Introduction to Vectors 46
3.2 Norm of a Vector; Vector Arithmetic 50
3.3 Dot Product; Projections 51
3.4 Cross Product 53
3.5 Lines and Planes in 3-Space 56
Review of Chapter 3 63
, iv
CHAPTER 4 COMPLEX NUMBERS 65
4.1 Complex numbers 66
4.2 Graphical representation 69
4.3 Equality of Complex numbers 74
4.4 Remarks 75
4.5 Polynominal equations 75
4.6 Polar form of a Complex number 77
APPENDIX Greek Alphabet 81
, v MAT1503
PREFACE
Introduction
Welcome to MAT1503, the first year mathematics module on linear algebra. Topics studied in linear
algebra, for example systems of linear equations and matrices, have a variety of applications in science,
engineering and industry. Amongst these are games of strategy, computer graphics, economic models,
forest management, cryptography, fractals, computed tomography, and a model for human hearing. So
as you can see there are a number of exciting applications of linear algebra. However, before you can
understand these applications you will have to familiarize yourself with the contents of this module.
MAT1503 prepares you for further studies in linear algebra at the second and third year levels. It also
equips you with the basic tools of linear algebra which can be applied in various fields.
Now in order to study this module successfully you need a good working knowledge of algebra at matric
level as well as the ability to think and work consistently.
Here are a few comments and advice from a colleague about thinking.
• Memorizing a theorem or a proof of a theorem, or memorizing an example or a method to do a
certain type of exercise, is not thinking. Thinking starts with questions like: What is this module
really about? What does this definition mean? What does Theorem x mean? Does Theorem x
surprise me or would I have expected something like it to hold? What are the ideas behind the proof
of Theorem x? If I tried to prove it myself without having yet seen a proof of it, how would I go
about it? Would I get stuck? Where and why? Could I get around this obstacle? And so on.
• Thinking is hard. It is much easier to pass certain modules by memorizing a few facts than it is to
think hard about and around the things you are studying. However, you do mathematics to make
the concepts involved part of your mental vocabulary and your worldview, not just to obtain a credit.
• One does not understand mathematical concepts, definitions and theorems after thinking about them
for 5 minutes. These concepts took hundreds of years to arrive at their current form. Mathematicians
spend thousands of hours thinking about mathematics, sometimes even about a particular problem
or concept!
• Do not be afraid to think. Thinking most often does not make you feel clever. On the contrary, it
is usually a slow, halting process which most often makes you realize exactly how little you know,
which usually makes you feel stupid and inadequate. You need to accept these feelings and think
about things anyway. Eventually, it will become easier and you will realize its value. It does not
, vi
matter if it takes a whole week/month/year to understand something. Once you have understood
it, it is part of you.
• Sitting writing at your desk is often not the best way to understand new ideas/theorems/definitions.
Read through the work and then mull over the ideas while you take a break/make tea/go outside.
It is often in these times that one understands things.
• Really understanding something is usually not done on the first attempt. Your mind does not work
that way. The best way to try to understand something difficult is firstly to ask yourself: What is
this about? Then try to find the main ideas first before going through all the details. Do not try to
force understanding, if after 15 minutes you have made no headway, take a break! But while you’re
on this break ask yourself where and why you’re getting stuck, exactly what it is that you don’t
understand. Also, it is very easy to get stuck in a certain pattern of thought about something. Try
to change your angle, as it were.
• Related to this is the fact that one does not study mathematics only by spending a fixed time each
day or week in front of your desk. The ideas in maths textbooks are alive, and you should adopt
them by thinking about and around them often, not just for an hour a day.
• If you think long enough and hard enough, you will eventually realize/understand/see something
that you did not before. It is then that you will realize how exciting and enriching thinking can be,
despite the fact that it can be so hard. (It can also become easy and fun, but only when you have
thought really hard about the concepts already!)
• Mathematics is primarily about ideas and concepts. In 10 years time you will remember no details
of any calculations done in this module but if you thought long and hard about the concepts, you
will remember what we were doing, and why.
• Working through countless exercises is useless without knowing what you are doing and why you are
doing it the way you are. We urge you to think about what you are doing at all times.
• You need to be brutally honest with yourself about what you do and do not understand. It is very
tempting to ignore the things you don’t understand and hope that they go away! Face these things,
no matter how long it takes. When you finally do understand them, it will be a great thrill!
• A good exercise to do is the following: After each section, explain to a friend who does not do
mathematics, what the section was about. If you cannot (that is, your friend does not understand)
then you may not yet understand the work.
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