Monday, March 26, 2007

Another 'erm' type of post...

... so you can't say that you weren't warned. If you're still 'eagerly' waiting for the posts that I keep saying that I'll post, then the only thing I have to say is that 'good things come to those who wait'. OK that sounds like a load of 'codswallop' but I couldn't think of anything better. You see I have three weeks of holidays, which if I'm lucky, will be filled by me trying to squeeze as much maths into the small thing that is known as a brain. So that's why I'm going to nicely space the posts out. They're not that big of deal, but they've got a topic unlike random posts like this!

Today was(is) a Monday. I had to state that because most of the times that I have holidays, I lose track of time, which isn't very good. I've got a feeling that the more determined I am to do some work, the less I'm going to get done. Call that Bean's 'amount of work done is inversely proportional to the square root of your determination' law or something like that. (Well Newton can have laws which sound cool, but make no sense whatsoever, (to me) so why can't I?!) The proof of that (if it's required) is obvious, and if you still reading that again with raised eyebrows then you'll know for sure in three weeks time. (BTW it wasn't really meant to make sense, but if it does then blame egm for linking this ! ~evil laugh~).

You've probably noticed that today wasn't a normal day, or that I actually might have done as the song suggested and beat my 'brain' against the wall. This wasn't so. It seems that I've gone into holiday mode again. It's only the first day of the holiday, but I went to sleep at some weird (read: early!) time. I previously commented on wanting to have approximately 8 hours of sleep, but another one of them weird laws, (which I shall not give a fancy name to ) is that the later you sleep, the later you wake up. Today I could blame the fact that I hadn't changed the time on my phone, so when I did wake up I thought I had another hour That's really a pathetic excuse, but I'm blaming the change in sleeping pattern for my headache, which I've had all day. (Or could that be due to the fact that I'm having withdrawal symptoms already, because I had no lectures today! aaah- what's going to happen to me in the summer holidays...!).

OK, enough of the dramatics (for now!), although I had a headache I decided to slowly ease myself into an ideal pattern for revision. If you're wondering whether I have anything else better to do with my time, then I don't blame you. I do do other stuff, but what I can say- I'm consumed by maths! (Not that I mind). So, obviously since I was taking things nice and slowly I chose to do the first chapter from the Linear Algebra book on vectors. Opening the book I was greeted with this:

"Where shall I begin, please your Majesty?" he asked. "
Begin at the beginning," the King said, gravely, "and go on till you come to the end: then stop."

-Lewis Caroll
Alice's Adventures in Wonderland. 1865

So that is what I hopefully intend to do. However, I don't see the 'end point' in my case, but I will assume that for Linear Algebra the end is the end of the book. Now I didn't get very far in this chapter, although I had hoped to, but I came across this neat proof. I have decided to post it here, since it's to do with maths(duh) and this also gives me another chance to mess around with LaTeX. Now at this moment in time, at the bottom of m
y screen it says 'could not connect to etc'. I'm taking it that my Internet connection is somehow not working, although my computer is telling me otherwise. Another reason for me to believe this is that I can't seem to access the AoPS site! So indeed my Internet has gone 'kaput', but I will finish typing this post up before heading off to bed. (Indeed in a very annoyed manner- my previous disjointness has now vanished, and boy am I mad!). Now, you may notice a change in my tone- so I apologise before hand for sounding like a sulky teenager (6 year old!) but I'm not too happy! :(

The other reason for including this proof (which took longer than anticipated to type up, because of no net) is to show that although it may be obvious, that u+v= v+ u (i.e. vector addition is commutative) but to prove this is another matter. The proof isn't too difficult, although I would never have thought of it myself, but I like how 'sim
ple' it is. Although I didn't have to use LaTeX, I did:Neat. May I also point out that writing this post at 00.00am and knowing that I will not post it for maybe a couple of days is making life difficult. I can't exactly go and wake my dad up and shout the Internet isn't working! He'll just reply that it might work in the morning or he'll have a look at it then. What am I then meant to do? Shout 'Dad, you don't understand, I have to post this post on my .... oops... erm no it's ok. We'll see what you can make of it in the morning! Night.' Sigh. It's funny how much I rely on the Internet! I don't exactly rely on it, but it's always there so to speak, and I'd gotten 'used' to 'surfing' it and checking various things out! Noddy seems to think that maybe it's not working for a good reason! Humbug :D

A short laugh surfaced for a second, whilst I remembered what initial random mumbo jumbo I was going to post! I might as well continue with that now, but if you're already asleep at this point or sick and tired of my moaning, then it's best you don't read on. I'm going to give somewhat of a boring history lesson, although it's more of a case of writing what's currently on my mind.(not sure whether to write this now, but I can always 'get rid of it' later I suppose!)

In year 8 (that was when I was about 13 I think), I noticed that I had back problems. I obviously went to the doctor, got a blood test done etc. and the Dr said that there was nothing wrong with me. I didn't really care at the time, but since my Dad has a bad back he was a bit worried. The problem continued to exist, but it didn't change my life in any way. I forgot about this and a few years passed by with me not caring too much and it was in year 11 (I think) when things changed. You see I acknowledged that my back was higgidly piggedly (new word!), but for some reason this made me more determined to overcome this fact. You could say I was stupid, and still am stupid as a matter of fact, but I deliberately pushed myself harder.

Same as when I sprained my ankle. That didn't stop me from playing football, although initially I did move very gingerly (it was painful!) but I felt that by playing with my bad ankle, it'd get better. There's no logic in that thought process, but even today as I go to uni I 'choose' to carry a back pack which weighs roughly 5-6kg. This is silly, but I don't really know why I do it. Maybe I could not take my water bottle, or leave a book at home, but I don't. There is one possible reason that I do this, but I can't put it into words. I guess it's a bit like people telling me to change my degree- that made me more determined to struggle on, and the same thing sort of applies to my back.

What's actually got me thinking about 'why' is that today, whilst I was doing some 'revision' I heard my brother playing cricket outside. My attention was slowly diverting from my books, and once the neighbours dog started to bark I realised that revision was pointless and headed outside. I've not played cricket in a while so I obviously decided to bat first. I was slowly getting into the swing of things, and after my brother had gotten me out three times I decided to let him bat. (I wasn't initially planning on bowling since my backs being a 'pain' :D ). Bowling is what I love doing in cricket. Batting is great when you want to vent your frustration, but if you don't bat 'calmly' more often then not you get out! You only get one chance, whereas with bowling you get more than one chance. I'm a medium/fast pacer, and like to pitch it up sometimes, but mostly I like to bowl a good length. (extra info if you're interested!)

It's pretty restrictive playing in the garden, but I bowled some loose deliveries to start of with. I knew that I shouldn't overdo it. I knew that I should continue bowling slowly to him. I knew this, yet I couldn't help myself and slowly started bowling slightly faster and faster. Now when you bowl you have to put your back into it, however like all other times, I don't realise that I'm overdoing it. You could say it's adrenaline, but even when I'd helped Milo move his stuff back into his university room in January, I didn't realise that I should 'chill out' for a second. It was painful then, as it is now. I've come to the conclusion that my back is funny shaped (or that I don't stand up straight all the time?), that is why I'm probably more susceptible to back problems. That's why I'm more determined to not let them affect me too much.

Ok I'm getting sidetracked here (again). I'm basically trying to say that although I haven't been nice to my back, but I adopt the same mentality in my maths and other aspects of my life! You could say that I see my back as a 'weakness' and so try to overcome it by using it to do what a normal person can do with their back, but slightly pushing it harder. Now in maths, this is obviously different, since when I face a difficulty and persist in trying to overcome it, the only harm it could do me is cause me great frustration and I could probably become hulk-like! I guess it's also because I don't like 'not' being able to do something, which I know that I can and should be able to do. (I could name a few maths topics here!) Weird!

Now this post is becoming longer than anticipated, but I guess the delay of no Internet is to blame for that. However, today I've realised that I'm 18 and playing this 'game' with my back is a serious game. If you ever get hurt take it easy. You see writing this down has me feeling very stupid indeed (I normally keep this to myself :o ). I know that after today's cricket game, tomorrow morning is going to be painful (that's why you should stick to football folks)! Lecture over. I hope I didn't scare you guys, and I don't get them everyday like it may seem. (At random times, like today, it plays up a bit, but apart from these times it's not too bad.)

Oh and I've calmed down after the Internet mess up- let's hope, for my sake(!) that's it not too long before it's working!


egm said...

Haha, now I'm to blame for stuff not making sense? Hmm! I've found a purpose in life after all!

Do take care of your back. See a doc about it if need be. And carry less books. I used to carry so much gunk in my backpack that I thought I'd need, and more often than not I barely used a number of them. And my back would hurt big time. So today I carry at most two books, (three if they are small paper backs). This has made a huge difference in how my back feels now.

As for your persistence in maths, it reminds me of something I read about Bernhard Riemann, and how he hated reading about something from the book. He would much rather work everything out for himself, discovering the various concepts on his own. This exasperated his teachers, but realising how skilled he was in maths, they bent the rules just for him in terms of handing in work. It didn't help that he was a perfectionist who wouldn't hand in work unless it was 100% correct.

egm said...

Here's a link to more on Riemann...

beans said...

Actually I've decided to blame you for everything that goes wrong! I've got to have someone, so that when my mum asks, 'Beans what the heck is this?!' I can reply, 'egm'. Haha, so we both have a use for this now. (I guess on this one instance it is your fault, since you put that physicy stuff in my head!):D

I know exactly what you mean about taking things because you think you might need them! In the end, you come home and wonder what was the need to take it. My problem is that I hate carrying stuff in my hands (most of the time) so I always shove stuff into my bag. My Doc is uselss, but I'm going to try to 'not take the extra books that I never use!'.

I never knew much about him, but he was a genius! We both may be persistant (in different ways of course), but I believe that I have to be, otherwise I'd never understand anything. (I'm too thick to discover stuff on my own ;) )

Jake said...

The proof that vector addition is commutative is somewhat tautological...

Most texts define a vector in terms of being an element of a vector space. The definition of a vector space states that the vectors under vector addition should form an abelian group so vector addition is commutative by definition.

The kind of proof you have would be best used the other way round e.g. given a set V of vectors and a field F; if one wanted to demonstrate that V formed a vector space over F then amongst the other requirements one would have to demonstrate that the vectors were commutative under the vector addition that you define.

Remember, vector addition could be anything you define it fact vectors need not even be expressible in terms of components.

beans said...

I think that's found later on in the book. The first chapter is just about vectors, and if I recall correctly, the 'theorem' was about vector properties.

But yes I see your point. Although I didn't know that vectors under additon from a vector space form an abelian group! (cool!)

Jake said...

Although I didn't know that vectors under additon from a vector space form an abelian group! (cool!)

Yes, given a set of vectors V and a field F, we normally say that V forms a vector space over F iff there is a vector addition defined such that V is an abelian group under it and there is a scalar multiplication F x V -> V such that scalar multiplication is distributive over vector and scalar addition and the multiplicative identity of F acts as an identity with respect to scalar multiplication i.e. (1.v = v for all v in V) and also, for all x,y in F (xy)v = x(yv)

In a lot of texts it just lists out the full list of criteria as eight seperate items but just remembering that the first four state that V is an abelian group under addition us easier to remember.

beans said...

Yes in my notes it's listed as eight (or was it 10!) properties. Little did I know, when I was initially taught about abelian groups, that they would prove to be useful elsewhere! Thanks. :)

(I kinda mocked them because of the fact that the 'a' is not captial- apologies!)

Jake said...

Yes in my notes it's listed as eight (or was it 10!) properties. Little did I know, when I was initially taught about abelian groups, that they would prove to be useful elsewhere! Thanks.

Yeah, group theory is not only very useful but also pretty fun. I think I am definately leaning more towards an interest in algebra than, say, analysis (although I have only really done some basic group theory and linear algebra and very little ring or field theory).

I have been following this blog recently which has quite a few posts on some elementary (and also some non-elementary!) group theory. I particularly enjoyed the posts on the group theoretical aspects of Rubik's cube.

beans said...

I don't really know what I'm leaning towards, but I know what I definitely don't want to be doing. (stats and mechanics have been crossed of my list!). Not really done much group theory, we only had 3ish lectures on it last semester, but it does seem 'interesting'. ~tiptoes~

I like the way it all links up. Like back in the section on polynomials we stated general results for collections
F [x] of polynomials with coefficients from the set F. Later after doing fields we were told that the symbol
F was used because we wanted the set of coefficients to be a field.

Analysis does seem 'interesting' although I'm not getting it at the moment. It's all about using what you know to help you get something else I suppose.

I do sometimes read that blog, and I like the guys rants! :D (will link it)

Jake said...

Not really done much group theory, we only had 3ish lectures on it last semester, but it does seem 'interesting'. ~tiptoes~

I like the way it all links up. Like back in the section on polynomials we stated general results for collections
F [x] of polynomials with coefficients from the set F. Later after doing fields we were told that the symbol
F was used because we wanted the set of coefficients to be a field.

Yeah, we haven't really done any group theory yet but I have done my own reading in the subject from Whitelaw's 'Abstract Algebra' and also Allenby's 'Groups, Rings and Fields' (this one is an absolute must read - brilliant book!)

beans said...

Hmmm, I have his name writen on a post it note with the title 'numbers and proofs' next to it. I'm assuming it's the same guy!

(It seems my reading list is growing! Thankfully I was today told that the library is open during the summer holidays, so there is still hope of actually reading some of the books).

The book that was recommended to me was 'Numbers, groups and codes' by Humphreys and Prest. Although I actually used that to help me with the modular arithmetic stuff, so can't really comment on how good it is on groups!

Jake said...

Although I actually used that to help me with the modular arithmetic stuff

In fact modular arithmetic is a good example of one of those things that links up with later studies in algebra. When you do group theory, you come across the concept of quotient groups (or factor groups) where the idea of modular arithmetic is generalised. To put it in crude terms; it is a way of forming a new group from an old one by sort of 'modding' out certain types of subgroups.

So for example you have
{Z,+}/{nZ,+} is isomorphic to {Z_n,+} (which is basically what modular arithmetic is i.e. disregarding multiples of n is the 'same' as working in Z_n

This concept then quickly leads to the first isomorphism theorem which basically says that given two groups G_1 and G_2 and a homomorphism g: G_1 -> G_2 that G_1/kernel(g) is isomorphic to image(g) whence the rank-nullity theorem of linear algebra becomes an obvious corollary.

....phew, sorry about that, got carried away!

beans said...

No don't apologise! It's sounds interesting. I keep hearing the words homo/isomorphism, kernal and a few others but never really knew what they're do with.

(At least I sort of followed the modular arithmetic bit and know the rank-nullity theorem from Linear Algebra. So there is some hope for me then!)

Thanks again. :)

Jake said...

I keep hearing the words homo/isomorphism, kernal and a few others but never really knew what they're do with

Well, homo/isomorphism are used in different contexts but normally mean loosely the same thing.

Given two groups (G,+) and (H,*)
A group homomorphism is a map f:G --> H such that
f(x+y) = f(x)*f(y) so in other words it preserves the group composition. A group isomorphism is a group homomorphism that is bijective; so in effect two isomorphic groups are pretty much exactly the same except for the naming of the elements.

The kernel of a homomorphism is the set {g in G: f(g)=1_H} where 1_H is the identity element of H. It turns out that the kernel of a homomorphism actually forms a normal subgroup of G.

beans said...

The weird thing is that rather than 'learning' what I should be, I now want to learn more about groups!

BTW which modules have you studied so far?

Rohit Jain said...

funny! :)