## Vectors and conics

It looks like the M208 honeymoon is finally over – I actually didn’t enjoy Unit LA1 very much at all. It was a bit of a strange unit, made up of an introduction to coordinate geometry, a couple of sections on vectors, and then a bit about conics. Perhaps I’m missing something, but there didn’t really seem to be much connection between the vector bits and the conics stuff, so it felt a bit weird to suddenly go from learning about dot products to messing about with parabolas.

I found it quite hard to get back into working with conics, even though they were covered in almost as much detail in MS221 last year; in fact, oddly enough, I had less trouble getting used to vectors again, and the last time I studied those was in 2007! I also made a complete mess of all the “draw the intersection of these two planes in $\mathbb{R}^3$” exercises – you’d think that drawing what are essentially two overlapping parallelograms would be fairly straightforward, but my sketches end up looking like Cubist interpretations of the planes. Hopefully there won’t be a sketching-planes-in-$\mathbb{R}^3$ question in the exam…

Anyway, I’m looking forward Unit LA2 – it’s about matrices, so hopefully it will be more my kind of thing!

## Cosets and Lagrange’s Theorem

Over the last couple of weeks I’ve been making my way cautiously through Unit GTA4 – I say cautiously because there are a couple of warnings in the text about how relatively difficult some students find the last section of this unit, so I wanted to make especially sure that I had a good understanding working with cosets before I tackled the intimidating Section 6.

Actually, I think I struggled more with Sections 2 and 3 than with the final section; there were a few questions that I just didn’t even know how to begin to answer (thankfully the solutions given at the end of the book did make sense when I eventually read them – so at least it wasn’t completely incomprehensible!).

Section 6 itself, which is about infinite quotient groups, was much more manageable than I expected – quite good fun, in fact! Earlier on in the unit, I was wowed by the fact that you can divide an infinite group, such as $(\mathbb{Z},+)$, into a finite number of cosets – I was even more impressed with the bit in Section 6 about dividing $\mathbb{R} / \mathbb{Z}$ into cosets that form an infinite quotient group. I’m not sure why exactly, but an infinite group having an underlying structure which is isomorphic to another infinite group pretty much blows my mind!

The question in TMA02 that relates to this unit was more straightforward than I expected; I think the questions based on GTA2 and GTA3 are a bit more challenging, although it could just be the fact that I’ve had a long time to get used to working with groups and subgroups now. So the first draft of TMA02 is now pretty much finished, bar any changes due to the feedback I’ll get from TMA01.

Now it’s on to the first unit of the Linear Algebra block, which is about vectors and conics. I must admit, I’m not particularly looking forward to this unit – I’d much rather get straight on with Group Theory Block B, but unfortunately I’ve got five units of linear algebra and four units of Analysis Block A to get through first!

## Permutations

I’m not a great fan of bank holidays, because I generally don’t like the disruption of my normal routines, but one thing bank holidays are great for is getting ahead with my study schedule! This week I’ve had loads of time to wrap up TMA01, and to work through Unit GTA3, which is about permutations.

I’d hoped that there would be some crossover between the combinatorics stuff that we covered in MS221 Unit B1, and the group theory permutations material in GTA3. It turned out that although there weren’t any explicit connections made between the two topics, an understanding of the basics of permutations (in the sense of questions like “how many different ways can I arrange the numbers 1, 2, 3 and 4?”) was definitely useful to have.