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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UCT MAM1000 lecture notes part 45 – 3D geometry and vectors part viii

We will discuss mostly three dimensions here, but what we have will be applicable to any number of dimensions (greater than or equal to 1). We want to be able to describe a straight line – a one dimensional object, infinitely long in both directions. We will see that vectors give us a perfect language with which to do this.

Remember that in three dimensions, a line can be defined by the intersection of two planes as in the intersection of the blue and the green planes defining the red line:

int

Each plane is specified by a single equation, and thus a line is specified by two equations (one for each plane). Here we will see that sometimes you just need one equation to specify a line, if you are using vectors, and sometimes it will seem that you need three equations, if you are using a parametric equation.

Let’s take a line, and specify some point on it.…

By | September 26th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|4 Comments
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UCT MAM1000 lecture notes part 44 – 3D geometry and vectors part vii

In the following, I’m going to miss out quite a few details which I think are very nicely laid out in Stewart. I will try and add a slightly more pedagogical tone to some of it, and some nice diagrams along the way.

So we saw in the last post that we can write the cross product of two vectors, which itself gives a vector, in terms of the determinant of a 3 by 3 array. We can use this to both find a vector perpendicular to two given vectors (unless they are parallel to one another) and also to find the area of a parallelogram formed by two vectors (the area of which is zero if the vectors are parallel to one another).

The second of these is easy enough to do in two dimensions, but in three dimensions that’s not an easy prospect. Using the cross  (otherwise called the vector) product makes this easy.…

By | September 25th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|3 Comments
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UCT MAM1000 lecture notes: More complex numbers practice

I’ve been asked a few times for more practice questions on complex numbers. This is where Wolfram Alpha can be your friend (like it’s not already!).

I’ll just give a few examples of questions from the tut on complex numbers which you could have solved using Wolfram Alpha, and from this you will be able to set up your own questions.

For instance, question 48 c) Find the roots of z^5=1+\sqrt(3) I can be solved in Wolfram Alpha with the command:

 

Solve[z^5==1+sqrt[3]I,z]

 

Moreover it will solve this for you, give you the five roots and plot them in the complex plane. So now you can come up with any root question you can possibly think of. There’s an infinite number of questions to start you off. You can thank me later!

If you want to convert between the trigonometric form and the exponential form, you can use the two commands:

 

TrigToExp[Sin[x]+2 I Cos[x]]

ExpToTrig[Exp[I z+3]]

 

Though remember the definition of the hypergeometric trig functions from a previous tut.…

By | September 20th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Undergraduate|0 Comments
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UCT MAM1000 lecture notes part 43 – 3D geometry and vectors part vi

Determinants

The idea of determinants have been about since around the 3rd century when it first appeared in an ancient Chinese book of Mathematics called The Nine Chapters on the Mathematical Art. It was used originally to define certain properties of systems of linear equations, as we will see later in the section on linear algebra, however for now we will simply use it as a particular way to easily calculate the cross product. Let’s take a two by two array of numbers and define the determinant for this.

 

\left|\begin{array}{cc}a & b \\ c & d \\\end{array}\right|=ad-bc

 

The vertical lines on the left and right are the sign that the we are taking a determinant. For now this is just a definition and we will work with it in what follows. Don’t worry too much about where it comes from, but we will see later where it comes from and we will see now why it is useful.…

By | September 19th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|6 Comments
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