Archive for February, 2010

Vector spaces

Longest. Unit. Ever. Or at least it feels like it, because I’ve been working through Unit LA3 on and off for about a month now! I was out of action with a cold for a couple of weeks in the middle, and when I came back to it I found that whatever understanding of vector spaces I had in the first place had completely disappeared. Perhaps I sneezed out the relevant brain cells…

I think the thing I had trouble with the most was spanning sets – particularly proving that a particular set spans a given vector space. I found the Vector Spaces chapter of Paul Dawkins’ excellent Linear Algebra class notes really helpful (though of course you’ve got to beware of the different notation used – I don’t want to incur the wrath of my tutor by using non-M208 notation!).

Linear independence took a bit to sink in, but I think I’ve just about got the hang of it, and bizarrely enough I found the later sections about orthogonal and orthonormal bases much easier than the earlier stuff. I’m quite intrigued by the remark at the end of the unit about orthogonality and polynomials – the idea of two polynomials being orthogonal to each other is very hard for me to get my head around, but I’d be interested in doing some more work on it. I wonder if it will turn up in any of the level 3 pure maths courses, or whether it’s more of an “applied” topic. The unit also mentioned that orthogonal polynomials are important in mathematical physics, so perhaps Alex will end up encountering them in one of his future courses.

In other news, I got my marked TMA01 Part 1 back last week, which came in at a nice 91%. I just hope I can manage the same kind of mark for Part 2. I didn’t get a great deal of feedback on Part 1 (which is understandable, since it’s only two questions), but my tutor did make a vague suggestion that I would be better off not word-processing my assignments in future. Unfortunately for her, I’ve already got TMA02 word-processed and printed out, ready to go! To be honest, I much prefer word-processing maths assignments – I hand-wrote all my TMAs for MST121, and had to scan or photocopy each one if I wanted a back-up copy in case the original got lost in the post. Very tedious indeed! So I’m not keen to go back to hand-writing assignments any time soon – they’ll have to pry my copy of MathType Lite from my cold, dead hands!

Now this is more like it! I wasn’t very impressed with unit LA1, but LA2 was much more my cup of tea. I enjoy working with matrices, especially using row reduction to solve systems of simultaneous equations. I love being able to quickly find a solution for what initially looks like an intimidating monster of a system, just by using the row reduction strategy. Unfortunately there’s lots of room for silly arithmetic errors when doing row-reduction, so I’m finding that I need to double-, triple- and quadruple-check my work at the moment.

I’m not sure why it never occurred to me before, but during LA2 I found myself suddenly wondering who had invented matrix methods for solving simultaneous equations, and I was really surprised to find that the technique has been known for around 2000 years! That’s pretty mindblowing, I think.

I was a bit disappointed with the shortness of the question on LA2 in TMA03 – the questions on LA1 were worth 20 marks altogether, and the question on LA3 is worth 25 marks, but the LA2 question is a relatively modest 10 marks. I was looking forward to a bit more of a substantial workout, but I guess that the matrix-related material is mostly just revision of topics from MS221 and MST121, so I can’t really complain that they didn’t spend enough time assessing it.

The next unit, LA3, is about vector spaces – which I know absolutely nothing about, so I’m really looking forward to it! Who knows, it could even be a topic that becomes one of my favourite areas of maths, like group theory suddenly did towards the end of MS221. That’s one of the things I love about studying, and particularly about studying maths; there’s so much wonderful stuff out there that I currently don’t even know exists, just waiting to be explored! 🙂