Polynomial division

Note: In the following I use the words power, and order somewhat interchangeably, in relation to the exponent of x.

Please also forgive the rather bad formatting for some of the expressions in this text. WordPress and LaTeX are somewhat unforgiving.

One of the topics which seems to have caused the most problems in the assignments for MAM1000W so far this year is that of polynomial division. Thus I want to go through an example here to show you exactly what we’re doing when we think about performing such a calculation.

If you think you already know what to do, but want to have some more practice, plug in some random polynomials into this page and make sure that you get the same steps as them.

In fact, the first question might be: what’s the point? Often the expression looks more complicated in the end than it does at the beginning.…

By | March 6th, 2016|Courses, First year, MAM1000, Undergraduate|8 Comments
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First year resources – part 4: revision

This is a continuation of the previous posts, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The following sections in the resource book are about mentors and the whiteboard workshop. They are really quite specific to the course, and more about the details than the philosophy of it, so I am not including them here.

The next section on the other hand is vital, as most students are never given much guidance in how best to revise, and it is one of the most important skills they can gain. I have written my thoughts from my own experience, but I am not trained specifically in education, so I am grateful for any additional thoughts.

 

How to revise

Revision is a bit like comedy: Timing is everything!

…
By | January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments
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First year resources – part 3: Tutorials

This is a continuation of the previous post, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The next part of the resource book is written by a former first year who very helpfully wrote “How I achieved over 80% in MAM1000W”. I am not including this here as I shan’t be altering what he has written

The next section is entitled “Tutorials”, and helps to make the tutorial problem sets and tutorial sessions themselves as useful as possible.

The weekly tutorial questions which you will be given to help practice what you’ve learnt in class are renowned for being time-intensive. This is true, but on top of going to lectures, these are the key element of the course for helping you to really master its content.

…
By | January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments
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First year resources – part 2: Why study mathematics?

This is a continuation of the previous post, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The second part of the resource book is a “Meet the team” section, which includes photos and a short bio of the convenor (me), the lecturers, and the senior tutors for the course.

The next section is entitled “Why study mathematics”, and is in some ways the most controversial/important section in here.

For quite a few of you, the immediate answer you may have thought of to this question is not one that will make you happy, but I hope that this section will give you reasons to feel really positive about taking this course.

For some of you, the reason to study maths is because you have to study it as a prerequisite for your course.

…
By | January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments
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First year resources – part 1: Intro to first year mathematics

As mentioned in the previous post, I am writing a resource book for the first year course that I will be in charge of this year. I want to crowd-source ideas for this a little. By that I mean that I will put up a number of the sections that I’m writing, and I’m keen to know, from those who have been through the course whether you feel that there is anything wrong, or missing from this. It is hard to know for someone who has not been on the learning end of a course like this for a long time what makes for the most useful information.

The second reason for doing this is that I think that these thoughts can be used for a wide variety of courses, and so if anyone wants to take these resources and use them in other contexts, then I will be very happy for them to do so.…

By | January 26th, 2016|Courses, First year, MAM1000, Undergraduate|3 Comments
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First year mathematics experience enhancement – a question for you!

I am coming to you today with questions. Well, questions based on some of my own ideas…

This year I will be not only teaching, but entirely in charge of the UCT first year mathematics for scientists courses, known as MAM1000W. I have a number of changes I plan on making, not so much to the syllabus, but to the extra activities associated with the course, in an attempt to make it as rich and deep a learning experience as I can.

The first step of this has been altering the structure of the resource book. The resource book is a PDF which will be sent to all first years taking the course. Historically, it contains a little about the course content, a bit about how your marks will be calculated, a bit about good practice in terms of how to work, and then the second half is filled with tutorial questions.…

By | January 25th, 2016|Courses, MAM1000|10 Comments
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UCT MAM1000 lecture notes part 50 – linear algebra part iii

Gauss reduction

So far we have seen that we have a way to translate a system of linear equations into a matrix. We can manipulate the matrix in ways which correspond to operations on the equations which keep the important information in the system of equations the same (ie. the solution of the equations before and after the operations is the same). We have seen a couple of examples of when we can read off the solution from the matrix having performed the operations. So far the order with which we perform the operations feels a bit arbitrary, although we know that we would like to get the matrix into reduced row echelon form. There is however a very systematic way of going about this, and the term for the process is called Gauss Reduction.

Here is a detailed view of what Gauss Reduction will give us:

Gauss Reduction:

To solve a system of linear equations:

 

1) First find the augmented coefficient matrix of the system of equations.…

By | October 4th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|1 Comment
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UCT MAM1000 lecture notes part 49 – linear algebra part ii

Matrices

Solving a system of linear equations is not technically difficult: just eliminate the variables in a systematic fashion. When there are only two or three variables, this is easy to manage. But for a bigger system, things can quickly get confusing. We need to develop a systematic method.

The first thing to notice is that the names of the variables don’t matter. Consider, for example, the two systems

 

\begin{array}{cc} x + y &=3\\ 2x-y &= 4 \end{array}

 

and

 

\begin{array}{cc} u + v &=3\\ 2u-v &= 4 \end{array}

 

It’s clear that if we ignore the names of the variables — x and y versus u and v — these two systems are the same. The reason we can tell that they’re the same is because the {\em coefficients} of the variables are the same and the numbers on the right hand side are the same. These are really the only things about a system of linear equations that matter, and so what we can do is strip the system down to its bare bones and rewrite it like this:

 

\left( \begin{array}{cc|c} 1&1&3\\ 2&-1&4 \end{array} \right)

 

This is an augmented coefficient matrix (in general, a rectangular array of numbers, like the above, is called a matrix; a matrix with an additional vertical line, which plays the same role as the equals signs in the original equations, is augmented).…

By | October 3rd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|1 Comment
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First year maths lecture notes subject links

Advice from former MAM1000W students

First semester

Some introductory notes about the course and random links

Introductory topics

MAM1019H notes

Proof methods

Functions, Continuity and limits

…
By | October 3rd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|3 Comments
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UCT MAM1000 lecture notes part 48 – linear algebra part i

These notes are taken from the resource book and were originally written by Dr Erwin. I will be editing and adding to them throughout. Most mistakes within them can thus be presumed to be mine rather than Dr Erwin’s.

In this section we are going to develop a new set of methods to solve a type of problem we are relatively familiar with. We will find a way to translate between methods we know well, but which turn out not to be very efficient, methods which are graphically very intuitive, but not very calculationally useful, and methods which are computationally extremely powerful, but appear rather abstract compared with the other two ways of looking at these problems. These three methods which we will utilise in detail in the coming sections are shown in the following diagram:

matrices.001

As we go through I will try and show how we can go between these apparently different formalisms.…

By | October 2nd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|1 Comment
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A project initiated at the AIMS-IMAGINARY workshop, November 2014, Cape Town | All Rights Reserved
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