I have just completed TMA03 on the linear algebra section of the course and I am just in the process of writing up the last question so that I can post it off in the next day or two. I worked on questions 5 and 6 on the train whilst travelling down to see my family in Norfolk. I always book the quiet coach but it isn't always that quiet. I am sometimes confused as to why parents with young children book this coach when they must know that their children are hardly going to be able to keep quiet!
I have enjoyed linear algebra. I wish that they had included another block of it in the course as they have done for group theory and analysis. As such, this was the first time in a few months that I have delved back into it and I seem to be able to manage the questions ok.
The first three questions were straight forward but the latter three required much more thought and were a nice challenge. One in particular on subspaces and dimensions really made me think about the material and I felt that it pushed the boundaries a bit more than some of the other questions. The question on quadrics was just a slog and it is one of those where a simple slip can mess up the whole answer.
I wonder how many students do this, but part of my process for answering the questions is to look for self-consistency in my answers. For example, if I diagonalise a symmetric matrix A, I make sure that PTAP really does lead to a diagonal matrix! It is a good way of making sure that my answers are likely to be correct.
Now that I have done a good chunk of the TMA's and I have had a bit of a break, I feel ready to go back to the books and start reading again. First off finish the last section of GTB3 on the counting theorem!
Thursday, 23 February 2012
Monday, 13 February 2012
TMA addiction
I am currently making good progress with the TMA's because I can't stop doing the questions. I keep thinking that I'll put the question paper away for a bit and get on with some more reading but I can't help having a little look at the next unanswered question. I tinker around with it and before I know it I am fleshing out the whole answer. So TMA02 is done and I will post it off today and I have already completed a couple of questions from TMA03.
I had the first part of TMA01 back today and I was pleased to get 35/35 but I am slightly worried that I might have messed up something in the equivalence relation question in part 2. I have been learning a bit more about equivalence relations from the discussions on the forum and there was some discussion about whether you needed to have the reflexive condition or not. I tried to think of examples where a relation is symmetric and transitive, but not reflexive, but I found it hard to do. The trick, apparently, is to have an element of a set that is not related to any other element, including itself. I think equivalence relations are whole topic in itself and I could spend hours getting bogged down in it (which I don't intend to do!).
I had the first part of TMA01 back today and I was pleased to get 35/35 but I am slightly worried that I might have messed up something in the equivalence relation question in part 2. I have been learning a bit more about equivalence relations from the discussions on the forum and there was some discussion about whether you needed to have the reflexive condition or not. I tried to think of examples where a relation is symmetric and transitive, but not reflexive, but I found it hard to do. The trick, apparently, is to have an element of a set that is not related to any other element, including itself. I think equivalence relations are whole topic in itself and I could spend hours getting bogged down in it (which I don't intend to do!).
Thursday, 9 February 2012
Sense of humour failure
I have had a slight sense of humour failure over M208 over the last week but I think I am past it now. Instead of working on the books at the end of the course I have been working through the TMAs and I have completed TMA01 and I have nearly finished TMA02.
The sense of humour failure occurred because I was getting frustrated with the language of the questions. The OU marks are very formalised and you have to tick all the boxes to get all of the marks but often it is a guessing game to decide how much to put in and how much to leave out of an answer. I want to be concise but on the other hand I don't want to miss marks for not showing enough working or not supporting an argument. It was doing my head in a bit.
In the end I have decided to say "stuff it"; I am going to answer the questions in a way which I think is best and most natural to me and I don't care if I don't tick all the boxes. Enough of this agonising!
Actually, I think the TMAs are very straight forward (unless I am missing something). I suppose that's because I have already worked to near the end of the course and I have a lot of familiarity with it, so some of the early stuff seems a doddle. Perhaps I am going to be in for a shock when I get my first TMA back! TMAs are also addictive. I like solving the puzzles and I will probably end up doing most of them before I knuckle down to the last four books of the course (I have just one more section in GTB3 to finish before I get onto the last four analysis books).
The sense of humour failure occurred because I was getting frustrated with the language of the questions. The OU marks are very formalised and you have to tick all the boxes to get all of the marks but often it is a guessing game to decide how much to put in and how much to leave out of an answer. I want to be concise but on the other hand I don't want to miss marks for not showing enough working or not supporting an argument. It was doing my head in a bit.
In the end I have decided to say "stuff it"; I am going to answer the questions in a way which I think is best and most natural to me and I don't care if I don't tick all the boxes. Enough of this agonising!
Actually, I think the TMAs are very straight forward (unless I am missing something). I suppose that's because I have already worked to near the end of the course and I have a lot of familiarity with it, so some of the early stuff seems a doddle. Perhaps I am going to be in for a shock when I get my first TMA back! TMAs are also addictive. I like solving the puzzles and I will probably end up doing most of them before I knuckle down to the last four books of the course (I have just one more section in GTB3 to finish before I get onto the last four analysis books).
Tuesday, 24 January 2012
We're off!
The course hasn't officially started yet, but it feels like it. The course website opened last Tuesday and I was finally able to get my hands on the first TMA. Since then, I feel like things have stepped up a gear and I am already having to devote more hours to my study. For starters, there is TMA01 part 1 to do and it was good revision. I did get myself in a bit of a twist to start with but things worked out ok in the end. It is amazing how one tiny slip can lead you down a blind alley. Then there is the sudden burst of activity on the forums to follow; a bit more revision of I1 to do; talking to my tutor about one of the problems I found in one of the exercises (that was far more interesting than I realised) and trying to keep up with my work with the books at the end of the course. Phew! I won't be able to keep this up at this rate. I may have to scale back to just doing the TMA's and the end of course work.
Other than that I have finished the chapter on Homomorphisms (GTB2) and have begun the chapter on Group Actions (GTB3). This is my final group theory book. I don't mind group theory too much and I find it ok to work with. In the chapter on Homomorphisms (I wish that word wasn't such a mouthful!) we find that they are a bit like Isomorphisms but with the strict one-one and onto condition relaxed. Homomorphisms are basically functions that map one group to another and in the mapping process some of the domain group properties are preserved. For example, the identity in the domain is mapped to the identity in the codomain, as are inverses and powers of elements. Also elements that are conjugate in the domain are also conjugate in the codomain.
The meat of the chapter comes with the discussion of Kernels and Images of a homomorphism. This is a bit like the discussion of Kernels and Images of a linear transformation in the Linear Algebra blocks. The Kernel of a homomorphism is the set of elements of the domain that map to the identity element in the codomain. The Image is the usual notion of an image of a function. It turns out that the Kernel is a normal subgroup of the domain group and the Image is a subgroup of the codomain group. The fact that the Kernel is a normal subgroup of the domain means, of course, that its cosets partition the domain and this leads on to the Correspondence Theorem, a return to Quotient Groups and finally the Isomorphism Theorem (I ain't going to discuss all that!). It's all a bit of a handful to remember. Reasonably straight forward at the time but instantly forgettable!
Other than that I have finished the chapter on Homomorphisms (GTB2) and have begun the chapter on Group Actions (GTB3). This is my final group theory book. I don't mind group theory too much and I find it ok to work with. In the chapter on Homomorphisms (I wish that word wasn't such a mouthful!) we find that they are a bit like Isomorphisms but with the strict one-one and onto condition relaxed. Homomorphisms are basically functions that map one group to another and in the mapping process some of the domain group properties are preserved. For example, the identity in the domain is mapped to the identity in the codomain, as are inverses and powers of elements. Also elements that are conjugate in the domain are also conjugate in the codomain.
The meat of the chapter comes with the discussion of Kernels and Images of a homomorphism. This is a bit like the discussion of Kernels and Images of a linear transformation in the Linear Algebra blocks. The Kernel of a homomorphism is the set of elements of the domain that map to the identity element in the codomain. The Image is the usual notion of an image of a function. It turns out that the Kernel is a normal subgroup of the domain group and the Image is a subgroup of the codomain group. The fact that the Kernel is a normal subgroup of the domain means, of course, that its cosets partition the domain and this leads on to the Correspondence Theorem, a return to Quotient Groups and finally the Isomorphism Theorem (I ain't going to discuss all that!). It's all a bit of a handful to remember. Reasonably straight forward at the time but instantly forgettable!
Monday, 16 January 2012
Starting to get course fatigue
I was starting to get course fatigue this week. I have been working on M208 since mid February last year and it is hard to sustain the momentum over that length of time. Each day I attempt to get through a page or two, but this week I found myself doing even less than that. The thought crept into mind that I just wanted to just get the last 5 books over with, so I could start consolidating what I have already learnt.
I think part of the problem with studying is that you are being led by the nose through a lot of material not all of which is going to be exciting. The tendency after a while is to think, "oh no, not another complex idea to grapple with" or "groan, not another proof to understand". I am a firm believer that you learn the most from exploring ideas on your own and there isn't much room for this on this course. It is all very much spoon feeding.
To entertain myself I was thinking about groups in general and one question that I came up with was what if you had the situation where all elements of a set were self-inverse under composition; could that set form a group or not? I had been thinking about finite groups of low order and imagining Cayley tables where the identity element is always along the leading diagonal.
I posed this question on the OUSA M208 forum but I quickly realised that it was not a question for the novice. It was also not easy to pose the question logically, anyway. A couple of theorems occurred to me. One was a corollary to Lagrange's theorem that if G is a group of prime order p, then G is a cyclic group. This is because the only numbers that can divide p are 1 and p and so the subgroups of G can only have order 1 or order p. These correspond to the subgroups {e} and G (e is the identity element of G). Moreover, as the only element with order 1 is e, the other elements in G must have order n, so the group G is cyclic.
So if the order of a finite group is prime then the group is cyclic. If we now consider these prime order groups where the order is greater than 2, then we can show that the elements of these groups cannot all be self-inverse. Why? Because to be a cyclic group we require that there is g element G of order n but the orders of g are all 2 (because they are self-inverse).
Hence, if a set contains p self-inverse elements and p is a prime number bigger than 2, then this set cannot form a group!
Interestingly, the sets with either one and two self-inverse elements can form groups under composition. {e} is a group and {e, g} can be a cyclic group since g can generate the group.
The other theorem which tackles this problem head on is Theorem 3.2 of GTA4. Let G be a group, with order greater than 2, in which each element except the identity has order 2. Then the order of G is a multiple of 4. So this means that groups of order 3 or more in which all the elements are self-inverse have to have an order of 4, 8, 12, 16...This agrees with what I deduced from the corollary to Lagrange's theorem, but is more stringent.
I think part of the problem with studying is that you are being led by the nose through a lot of material not all of which is going to be exciting. The tendency after a while is to think, "oh no, not another complex idea to grapple with" or "groan, not another proof to understand". I am a firm believer that you learn the most from exploring ideas on your own and there isn't much room for this on this course. It is all very much spoon feeding.
To entertain myself I was thinking about groups in general and one question that I came up with was what if you had the situation where all elements of a set were self-inverse under composition; could that set form a group or not? I had been thinking about finite groups of low order and imagining Cayley tables where the identity element is always along the leading diagonal.
I posed this question on the OUSA M208 forum but I quickly realised that it was not a question for the novice. It was also not easy to pose the question logically, anyway. A couple of theorems occurred to me. One was a corollary to Lagrange's theorem that if G is a group of prime order p, then G is a cyclic group. This is because the only numbers that can divide p are 1 and p and so the subgroups of G can only have order 1 or order p. These correspond to the subgroups {e} and G (e is the identity element of G). Moreover, as the only element with order 1 is e, the other elements in G must have order n, so the group G is cyclic.
So if the order of a finite group is prime then the group is cyclic. If we now consider these prime order groups where the order is greater than 2, then we can show that the elements of these groups cannot all be self-inverse. Why? Because to be a cyclic group we require that there is g element G of order n but the orders of g are all 2 (because they are self-inverse).
Hence, if a set contains p self-inverse elements and p is a prime number bigger than 2, then this set cannot form a group!
Interestingly, the sets with either one and two self-inverse elements can form groups under composition. {e} is a group and {e, g} can be a cyclic group since g can generate the group.
The other theorem which tackles this problem head on is Theorem 3.2 of GTA4. Let G be a group, with order greater than 2, in which each element except the identity has order 2. Then the order of G is a multiple of 4. So this means that groups of order 3 or more in which all the elements are self-inverse have to have an order of 4, 8, 12, 16...This agrees with what I deduced from the corollary to Lagrange's theorem, but is more stringent.
Tuesday, 27 December 2011
Are things getting tougher?
I have now been working through GTB1 of M208 for the last three weeks and I still haven't finished it. I am beginning to wonder if the last two blocks of this course are tougher than the rest. I know that Christmas has slowed things down a bit, but usually I can complete a book in three weeks. One question was a corker. In Exercise 4.7 on p44, you are asked to find all the normal symmetry subgroups of regular hexagon. It sounds innocuous enough, but I ended up covering four sides of A4 with my answer.
GTB1 starts with some much needed revision of Group Theory. It is amazing what you forget when you have head full of analysis. Still, there are some things that are beginning to stick in my leaky brain.
The rest of the chapter starts to delve more deeply into conjugacy and normal subgroups. One essential idea is that conjugacy in symmetry groups represents symmetries that have a similar type. For example, when considering the square, the two reflectional symmetries which are associated with the lines of symmetry that pass through the corners of the square are of the same geometric type and are related by conjugacy. This is built up into the Fixed Point Theorem later in the chapter.
Another major section of the chapter is an exploration of the relationship between conjugacy and normal subgroups and this leads to four properties of subgroups that characterise normality. In the final section there is a look at infinite groups of 2x2 matrices.
I must say that it is tricky stuff. I find it hard to get all these abstract ideas ordered someway in my head so that they can be remembered. There is layer upon layer of ideas and whilst it is ok answering questions when you have just completed a topic, I can imagine that in an exam, it may be hard to come up with the right techniques for answering a random question.
Still, I am very glad that I have done so much advance work on this course as I will be able to spend time mastering the methods when the course actually starts. I am very much looking forward to my first assessment.
GTB1 starts with some much needed revision of Group Theory. It is amazing what you forget when you have head full of analysis. Still, there are some things that are beginning to stick in my leaky brain.
The rest of the chapter starts to delve more deeply into conjugacy and normal subgroups. One essential idea is that conjugacy in symmetry groups represents symmetries that have a similar type. For example, when considering the square, the two reflectional symmetries which are associated with the lines of symmetry that pass through the corners of the square are of the same geometric type and are related by conjugacy. This is built up into the Fixed Point Theorem later in the chapter.
Another major section of the chapter is an exploration of the relationship between conjugacy and normal subgroups and this leads to four properties of subgroups that characterise normality. In the final section there is a look at infinite groups of 2x2 matrices.
I must say that it is tricky stuff. I find it hard to get all these abstract ideas ordered someway in my head so that they can be remembered. There is layer upon layer of ideas and whilst it is ok answering questions when you have just completed a topic, I can imagine that in an exam, it may be hard to come up with the right techniques for answering a random question.
Still, I am very glad that I have done so much advance work on this course as I will be able to spend time mastering the methods when the course actually starts. I am very much looking forward to my first assessment.
Monday, 12 December 2011
End of Analysis Block A
I have now completed the last of the books in the first analysis block of M208. This book was all about continuity in real functions, and as for the other analysis books, I really enjoyed working on it. In a nutshell, to determine whether a function f(x) is continuous at x=a, you have to ensure that, as x tends to a, f(x) tends to f(a). Of course, you need to consider the approach to x=a from both smaller and larger values of x.
I am already familiar with the idea of continuity having come across it in previous maths studies but this course deals with the subject in a much more rigorous way, using sequences in x to determine how f(x) behaves as x tends to a. One gripe that I have is that a major plank of the rest of the book, the Intermediate Value Theorem, is only proved for a special case and not proved in general. When a lot of M208 is devoted to carefully proving theorems it is sometimes surprising to find omissions, but I suppose that in some cases the proofs are too involved to reproduce for the course. The Intermediate Value Theorem is the basis of how we can define inverse functions for increasing or decreasing continuous functions.
Another gripe that I have is that it is sometimes difficult to judge the level of detail that is required in some of the answers to the exercises. Sometimes I find myself putting in too much detail and at others too little. Take Ex. 4.1 on p38 of AA4, for example. The solution says that f(n)=n2-1/n tends to infinity as n tends to infinity by the Reciprocal Rule. Is it really necessary to quote the Reciprocal Rule at this point? Isn't this obvious enough? In the examples in the text they manage fine without quoting this rule, so why suddenly do so here? There has to be some point at which you don't have to quote every theorem and rule that you have learned in M208 in order to prove something otherwise it is going to be extremely tedious/arduous. In fact, the solutions to the exercises are meant to be a guide as to how questions should be answered, but some rules, strategies and theorems are omitted in these solutions when convenient. So how are we to judge this?
I am already familiar with the idea of continuity having come across it in previous maths studies but this course deals with the subject in a much more rigorous way, using sequences in x to determine how f(x) behaves as x tends to a. One gripe that I have is that a major plank of the rest of the book, the Intermediate Value Theorem, is only proved for a special case and not proved in general. When a lot of M208 is devoted to carefully proving theorems it is sometimes surprising to find omissions, but I suppose that in some cases the proofs are too involved to reproduce for the course. The Intermediate Value Theorem is the basis of how we can define inverse functions for increasing or decreasing continuous functions.
Another gripe that I have is that it is sometimes difficult to judge the level of detail that is required in some of the answers to the exercises. Sometimes I find myself putting in too much detail and at others too little. Take Ex. 4.1 on p38 of AA4, for example. The solution says that f(n)=n2-1/n tends to infinity as n tends to infinity by the Reciprocal Rule. Is it really necessary to quote the Reciprocal Rule at this point? Isn't this obvious enough? In the examples in the text they manage fine without quoting this rule, so why suddenly do so here? There has to be some point at which you don't have to quote every theorem and rule that you have learned in M208 in order to prove something otherwise it is going to be extremely tedious/arduous. In fact, the solutions to the exercises are meant to be a guide as to how questions should be answered, but some rules, strategies and theorems are omitted in these solutions when convenient. So how are we to judge this?
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