## Thursday, April 19, 2007

### Air of a distracted person..

Or so I was told today. Yes, I confess I seemed very distracted today and funnily enough the reason being maths. Now this week has been pretty hectic in terms of course works etc. I bounced some and then slowly the bouncing stopped and started again etc. I have mentioned before that I am somewhat of a free spirit. Sometimes I like going of on my own and doing whatever I do (maybe nothing) but having no care in the world about things and people. You know it may seem otherwise but I can appreciate silence (i.e. mulling over some maths!) and nature at times- and today seemed one of them days. I had also previously decided that in any free spare time that I have in uni, I'll try to get work done. Two reasons for this- one being that at home I probably won't get the work done, and secondly I need to start concrete revision so shouldn't do the home works at home.

In terms of this plan I did pretty good. In the morning I did a sequence and series problem sheet, and the graph of course. I also now have a copy of this book:
Nope, I didn't go and buy it- that would have required ordering it, which would have taken a while. Call me eager, but I got it from the library this morning and it's a bit battered but looks interesting. (I read the first page and the like the sound of the guy who wrote it). Unfortunately since I'd planned to do some sequence and series work I'd taken a book to uni with me. I don't like carrying stuff in my hands- that's what my back packs for, so once again I had a heavy load. Anyway, it seems that I under estimate how long it'll take me to get from the mss building to the Chemistry one. Once again I was pushing the limit but made it to the lecture exactly on time.

Now I have some advice if any one's interested. If your lecturer ever writes the following on the board don't do what I did!

Theorem:
Let V be a k-space with dimension n (dim V=n). Then,
a) Any linearly independent set in V contains at most n vectors
b) Any spanning set contains at least n vectors
c) Any linearly independent set of exactly n vectors is a basis
d) Any spanning set of exactly n vectors is a basis.

Now as I was writing these down, (d) struck me as weird and made me think 'eh'. a,b and c made sense but I couldn't see (d). So I waited for the proof. However the lecturer gave us an offer. He asked us which one he'd like us to prove- I said number 3. Unconsciously of course. So if you want to kill me and have three bottles one of which has poison in it, then make sure to put the poison in the third bottle since that's the one I'd choose! (but then again if you were to offer me three bottles I'd know you're trying to kill me so would kindly decline- ha!). Thankfully I didn't say this too loudly and I wasn't heard. Phew. So the lecturer was obviously met with silence and then Prof S asked us to shout any letter out. My stupid brain then realised that oops 3 is not a letter and since (c) is the third letter of the alphabet I said c. Loud enough to be heard!

The lecturer said (not to me) that it was the best choice. I then looked down on the board and spotted (d) and thought crap. Especially since the proof of (c) although I wouldn't have been able to do it myself, was shortish and 'simple'. Mull over (d) I will, but I can't help but think I've done a stupid thing. :o Please tell me there's another reason why (c) was a good choice... please! So children of all ages. Choose anything but (c). Choose anything but the letter associated with your favorite number. Don't be rash- think before you open your mouth!

Hmmm, work on a,b, d I must (in a while). However today I must confess that Bella and Panya would not stop talking (well they had long conversations, so did pause at times but you get what I mean). You see the air of a distracted person is that of a 'focused' person so to speak. Now in lectures it's sometimes easy to start day dreaming, however whats important is how quickly you snap out of it. Having people talk makes me angry, so I tend to focus my energy on this rather than trying to strain to listen. Sigh. I seem to have a thing for this and continuously mention it, but if you went shopping yesterday, or met a long lost friend then can't you tell someone after the bloody lecture? The tweenies know that I don't like talking so they generally don't talk to me during lectures, but Panya and Bella seem to talk throughout nearly every lecture. (normally this doesn't bother me since I don't sit next to them, but alas today I did.)

After this lecture, Fizz and Milo wanted to do Matlab so I went with them so I could do my homework for tomorrow. I'm confused because we're no longer using 'vectors' and it also seems that I have forgotten how to find a basis! I mean the question just asks for a basis of the subspace, so can I choose whether to find a basis of the 'column' space, null space or row space? Sigh once again. After this we had the sequence and series lecture. Giving up counting how many people turn up but we started at 2:05 and finished at 2:35pm. I'm probably the only stinking person who seems to have something against this! I mean today the Linear Algebra ran over for 30 second :D(not that I cared) but I wouldn't have minded being taught linear algebra at least in them 15 odd minutes. (However something clicked today- it was the smallest of things, but that's what kept the bad vibes away so I'm happy!)

We got our sequences and series coursework back today and I've done alright in it. Once again the issue of broadcasting results pains my brain. I mean everyone went to the Newman building to get the course works, so it wasn't just us. Now I told my marks to the tweenies, but I didn't want to tell them in front of everyone :\ Weird I know, but I got my script, dodged the questions and went to a side. When it was just us there I told them my marks. End of story. Why would I want the whole of the first years knowing my marks. They probably don't give a damn, but I don't know... like I said before- the moment people view you as a 'grade' things change. Especially when it concerns students! With teachers this isn't a problem. I can't seem to write what I want. We all did good and that was that- I'm actually quite eager to get the marks for the calculus test because I want to know how the question that I had a problem on is meant to be done. /sad

It seems no one has an answer regarding my question of whether the 'theorem' is a theorem. Has anyone actually attempted the exercise below? I mean exercise are meant to be done right? You don't want me to start crying do you! Would you shoot me if I said it's trivial? :D Ha well I want to give a clue, but then some of you might actually shoot me!

So I'm distracted because I'm focused. Does this mean that the remaining three weeks are going to be miserable? I don't think so. I mean I thought that I'd be posting less, but it seems that I should have done that in the holidays. You see it's less worrying to post after having done some work, since then I feel that I'm not wasting my 'time'. My 'distracted self' just means that I'm more disjoint than normal and will be more bouncy. I can't remember A level mechanics, but do I say the 'coefficient of restitution is low'? Or something like that e=0 (or is that to do with elasticity?). Anyway, if you do see me around, I'll probably be clutching my arm having walked into the lamp post :D (Don't be frightened, being disjoint feels quite good actually- everything apart from maths seems like background music.)

Steve said...

Please tell me there's another reason why (c) was a good choice... please!

Because (d) is easy to prove and follows from (b). Suppose S is a spanning set with n vectors. All you need to do is show S is linearly independent and then you're done.
OK, suppose it isn't, so a_1v_1+...+a_nv_n=0 where not all the a_i's are zero and S={v_1,...,v_n}. Suppose wlog* a_1 is not 0 then can you see that n-1 of the vectors in S span V? Can you see how you've then proved (d)?

*I assume you know wlog means "without loss of generality"?

beans said...

so n-1 vectors in S, span V. However by (b) this can't be possible since any spanning set contains at least n vectors. So our assumption is false- S is linearly independent and so a basis?

(Yep, I know it means that. :D That's another one of my favorite abbreviations! Sweet. Even better when you use it in day to day things!:o )

Steve said...

That's it, though you also need to say which n-1 vectors span V and why, and how have you used a_1 is non-zero, even if it is blindingly obvious to you.

beans said...

Thanks, will do. :)

So (c) might have been a good choice after all. :D Makes me feel relatively better!