Friday, April 20, 2007

Still awake

In my sleep deprived state I manage to fight of any other demons to write this post. I was doing my linear algebra homework a while ago. After completing part 1i, I became slightly stuck on the rest of the questions. I can't seem to see what to do, I'm meant to find a basis for the subspaces given but they're for nxn matrices. Anyway, recently the work that I've been handing in hasn't been of a desirable standard. You see my supervisor has me writing proper sentences- I'm eternally grateful for this, although in today's lecture I didn't write three dots for therefore for the first time ever! Shocking!

I know that my supervisor puts effort into marking (we get 'empty' grades- empty in the sense they're not technically recorded), and I have come to appreciate this effort. I know we're expected to pick up the skills of the game as we progress however before playing a new game, it's always wise to play the tutorial level. This shows you how to use your weapons and how you can manoeuvre through the game successfully. My supervisions have achieved this and I really do cry inside to learn that we won't be having them next year.

Anyway, I decided since I wasn't able to do question1 I'd attempt the proof question which also had to be handed in! Now I'm not very good at proving things, but what matters is that I at least attempt the question.

The question was:

6. Let V be K-space and S a spanning set such that every v in V has a unique representation as a linear combination of the elements of S. Show that S is a basis.

Now this probably seems trivial to a few people, however not to me. (Oh something else funny happened in the lecture today- Prof S said that something was 'obvious'. A smile formed on my face because I obviously didn't think so, and then he went on to ask whether anyone didn't find it obvious! I couldn't help but smile and think to myself 'me I don't' but obviously kept quiet. I probably looked like a looney, smiling to myself, but I was grateful that he went on to explain why it was obvious. :o )

So firstly I looked through my notes for anything similar. Didn't find anything but decided to see what I could do. Now I can ask you to be nice and not point out any obvious mistake that I've made- but how the heck will that help me!! I managed to muster something together which is the reason of this post. I feel pretty cool about it, but that isn't going to last for long of course! So what do you reckon? I think that it's not correct since I seemed to have used random mumbo jumbo stuff from my brain. It seems to good to be true! Have fun. :D
It's not very neat, but it looks beautiful to me (for now anyway). This maybe proof will explain tomorrows 'disjointness' indeed.

(sorry about the picture-its been taken from a phone so it's not very good)

Steve said...

That's very much on the right lines but you can shorten the proof a lot more towards Prof S's 'obvious'.

You have the line d_1v_1+...d_nv_n=0 with one of the d's not being zero. That's all you need to do! Why? Because there's a different very obvious way to write 0 as a linear combination of elements of S ... (to be continued by beans :))

beans said...

Haha- during the Linear Algebra lecture today, as I thought of my homework, I realised this as well! (Well I though that maybe I could use the theroem he'd given).

... will be continued later! (I'm a bit brain dead at the moment :o )