Pointers for exam preparation

The purpose of this page is for the teaching assistants to give some advice on how to prepare for the exam in TMA4145. This course is, for many of you, notably different from the ones you have taken before in that you are asked not only to compute or memorise, but also to rigorously prove mathematical statements. Since the last part is where most people have had problems, it will be our focal point.

The first part consists of common mistakes, misconceptions, or omissions that the teaching assistants encountered in supervision sessions or whilst grading the submitted exercise sets.

The second part is more directly relevant for the exam preparations, including tips on how to approach the problems given on the exam day.

Without doubt the most common source of any error is not having a firm grasp of the definitions involved – you cannot solve any exercise without knowing precisely what the mathematical objects you are working with are. Additionally, when there are several equivalent definitions available, make sure to pick a definition which suits the problem at hand.

Another frequent omission when using theorems is not checking that it applies, that is, verifying that all hypotheses in the statement of the theorem hold. A particular, and recurrent, example is the Banach fixed point theorem, which can only hold in complete and non-empty metric spaces.

Here are some more specific mistakes or misconceptions (in no particular order):

• being imprecise with notation, especially when it comes to elements and sets.
• choosing bad notation, particularly when working with sequences of sequences.
• mixing \(\max\) and \(\sup\) – the former may not always exists. Moreover, not specifying what set you are taking \(\sup\) over.
• when asked to compute operator norms, only giving bounds on them.
• the notion of bounded/unbounded linear transformations (vs. boundedness for functions in general)
• not knowing the properties of the different sequence spaces.
• the different characterisations of a boundary and of closed sets.
• proving only one of the implications of an equivalence (if and only if-statement).
• for Jordan normal forms, not matching the ordering of the eigenvalues and the Jordan chains.

This section gives suggestions on how to develop the abilities needed to eliminate the problems described above, the most important being the ability to write mathematics in a precise way.

A sensible way of learning the material in this course is the following.

• First, gather a certain familiarity with the lectures notes, not necessarily learning everything by heart, but at least to the point of knowing where to look if you are stuck.
• Second, redo the exercises from the exercise sets (these form part of the curriculum); whenever you are uncertain how to proceed, make sure you know all the terms involved. When you have struggled enough with an exercise to appreciate a solution, and only then, have look at the solution set – but try thinking about how you could potentially have come up with the solution by yourself and if not, why.
• Third, have a look at previous exams, they will give an indication of what is expected, but remember that the course material have changed over the years.

When doing exercises from the exercise sets, even the ones you have done before, convince yourself that you understand why every step you take is valid, and that you are able to convey this understanding in writing. This is not easy, but practising on problems you have already understood is a good way of learning it.

Having done this, working through examples given in the lecture notes – without looking – then comparing what you have written to the notes, can give you an indication of what you need to work on; this is also a good way of repeating the material.

When you feel you can accurately get across simple mathematical ideas, you could try to gain a deeper understanding of what is going on. A good way of doing this is following the advice from the famous Paul R. Halmos:

Don't just read it; fight it! 
Ask your own questions, look for your own examples, discover your own proofs. 
Is the hypothesis necessary? Is the converse true? 
What happens in the classical special case? What about the degenerate cases? 
Where does the proof use the hypothesis?

Some practical advice for the day of the exam is maybe also pertinent.

First, the required exam advice: Before attempting any of the problems, read through the entire exam and determine how much time you can spend on each problem, and which, if not all, of them you are going to try to solve.

When answering an exam questions, you are trying to convince the reader, in this case the grader, that you understood what you are doing. A good way of doing this is letting him or her in on the thought process; this way, you may also get partial credit in the event you are not able to complete the problem. Conversely, if all you write is a string of non-justified implications, some of which may be wrong, it is difficult to accurately assess how much you have really understood.

If asked to come up with examples or counter-examples, try the easiest examples you can come up with first. If this does not work, try modifying examples you are already familiar with (here it helps to have a good awareness of the examples used in lectures and solutions sets).