Category: M208: Pure mathematics


So it turns out that GTB2 (Homomorphisms) is quite a tough one as well! Wonderful stuff, just very taxing, especially in Section 4. I’m not sure why, but I had the worst trouble with exercises involving groups of complex numbers. It’s bizarre, because I don’t remember having much of a problem with complex number generally in the past, but having to find the kernel and images of isomorphisms from \mathbb{C} to \mathbb{C} seems to really trip me up. And even worse, the associated TMA question was full of that stuff! Hopefully there won’t be as many \mathbb{C} related questions on the exam…

GTB3, the unit about group actions, was quite gruelling too. I struggled like crazy with the exercises about group actions on 3D objects like cubes, I just seem to have a no knack for visualising and mentally “rotating” the things! And that bit in the DVD segment about rings of coloured beads and chessboards had my head spinning. I still find group theory one of the most interesting topics in M208, and in maths in general, but I just wish I was a bit better at it!

Weirdly, I’ll be really quite glad when M208 is finished, which is something I absolutely couldn’t have imagined myself saying six months ago. Even though I’ve had slightly heavier course loads in the past (in terms of points/credits), I feel like this year has demanded the most actual effort, and to be honest I feel intellectually knackered! So once M208 is over and done with, I’ve decided to take a year or two out to recharge my batteries. Ideally I’d like to pick up my studies again once the maths courses have been shifted over to autumn starts (assuming that still goes ahead), as that kind of timeframe would be much more convenient for me. Or I’ll just buy the M381/M336 textbooks from the OUW shop and study them at my own pace. Probably the latter since it’s the cheapest option, given how financially tight things are likely to be for the next few years!

Conjugacy

Wow, GTB1 Conjugacy was a lot more gruelling than I expected! I really thought that I’d sail through this unit, given how much I enjoyed Group Theory Block A, but I actually thought the middle sections of the unit contained some of the toughest material I’ve encountered so far. Perhaps I’m just very out of practice, but it really did seem like a step up in complexity – and in hindsight, I’m quite happy about that, because now that I’ve finished the unit I feel like I’ve had a little taste of more intense and esoteric group theory work. And I’m still interested in doing M336: Groups and geometry at some point in the future, so this unit can’t have been that bad!

I struggled a bit with the associated TMA questions, though. I tend to find anything involving visualising the symmetries of three-dimensional shapes quite difficult, and without giving too much away about that particular question, I can safely say that TMA05 gave me quite a workout in that respect. I also fell foul of my usual “misreading the question, and consequently trying to solve a much harder problem than it actually is” issue – I spent about an hour bashing my head against a particular part of Question 1, only to realise that I’d omitted one important word from my reading of the question, which meant that I could prove the required result in about 10 minutes. Arrgh! I hope I can manage a more careful reading of the exam questions than I seem to do for TMA questions!

So now I’m making a start on GTB2 Homomorphisms, which seems a bit more straightforward than GTB1 – at least so far! Who knows what lurks in the later sections…

Continuity

I finally finished Analysis Block A today, and oddly enough I’m a bit sad to see it go. Unit AA4: Continuity was a lot more fun that I expected, and I particularly enjoyed the section about the Intermediate Value Theorem. The bit in the DVD programme about antipodal points blew my mind – I was convinced that it was impossible to prove that you can always find two antipodal points on the Earth’s equator that have the same temperature, right up until they explained why it must be true. I even made Alex come over and watch that section, it impressed me that much! (The similar examples involving balancing a table on bumpy ground, and equally distributing sugar on pancakes, were lots of fun too.)

TMA04 is wrapped up now, and I’m proud to say that this will be my first ever LaTex-produced assignment! Well, I should say Lyx-produced, because no matter how much I tried to get to grips with LaTex editors like TeXnicCenter, I just couldn’t get the hang of it, so I ended up resorting to a more WYSIWYG-style program instead.

I really enjoyed using Lyx to type up this assignment, so I reckon I’ll be sticking with this approach in future (though I wouldn’t rule out going back to TeXnicCenter or a similar editor – perhaps when I’ve gotten thoroughly used to Lyx itself). And I’d definitely recommend Lyx for anyone who’s interested in moving away from word-processing assignments; I think Lyx has just enough WYSIWYG elements to make it a relatively easy program to learn to use.

So now, at last, I can finally get on with Group Theory Block B – it’s been a long time, so let’s hope I haven’t forgotten everything from Block A!

Sequences & Series

I found myself a bit underwhelmed by Unit AA2, to be honest. It was interesting, and I really enjoyed the part of the associated DVD program about approximating pi by taking exterior and interior polygons around a circle, but overall the unit didn’t quite grab me in the same way as the chapter in MS221 about sequences.

Unit AA3 was much more satisfying, and in a way it felt like AA2 was really just an introduction to the ideas we needed for AA3. For me, working with limits feels a lot more rewarding when it’s part of an attempt to prove that an infinite series converges/diverges. And the behaviour of infinite series is quite an interesting topic in itself – especially counter-intuitive results like the fact that \sum_{n=0}^{\infty } 1/n^2 is convergent but \sum_{n=0}^{\infty } 1/n is divergent. I really do love results that make me go “but.. but how??” like that.

Still, even though AA3 was more enjoyable than AA2, I think the Analysis Block A has been a bit of a let-down compared to the first half of the course. I’m quite looking forward to unit AA4, as it’s about continuity, and I’m quite eager to find out how on earth continuity can be formally defined (since the only definition I’ve ever come across is the informal “you can draw the graph without picking your pen up” one), but my main aim at the moment is to make it through Analysis Block A so that I can enjoy the sweet, sweet goodness of Group Theory Block B. There are only three units of it, but I’m going to savour every page!

Numbers

I finally finished the TMA question associated with Unit AA1: Numbers today, and it’s quite a relief! I feel like I’ve been working on AA1 for absolutely ages. It’s not a bad unit, by any means, but for some reason my motivation levels have been a lot lower than usual lately, so I’ve barely been putting any time into M208 for the last few weeks.

The trickiest bit of AA1, for me, was the whole Proving Inequalities section. I love proof by induction, I really do, but I seemed to have a great deal of trouble with all the exercises involving it in AA1. I’ve also realised that if the exercise doesn’t explicitly state which method of proof I should use, I generally opt for entirely the wrong choice. I spent ages agonising over a particular problem that I was trying to solve using induction, and then realised when I finally relented and re-read the appropriate bit of the unit, that it was actually possible to give a direct proof fairly easily. I’m pretty good at over-complicating things, it seems!

I got TMA02 back through the post this week, which is probably what gave me the impetus I needed to finally finish this unit and the TMA question about it. I managed 92% on TMA02, which I’m pretty much okay with – I wouldn’t have been surprised if it had come back much lower than that, since I generally feel like a bit of an imposter on this course.

Anyway, now it’s time to make a start on Unit AA2: Sequences. I remember really enjoying the bits about sequences in MS221, so hopefully this unit won’t take me another three weeks to work through!

Eigenvectors

Hyperboloid by fdecomite

I finally finished the Linear Algebra block this week, which I’m very happy about. It’s not that I didn’t enjoy it, but I feel like I’ve been working through it for years, not months!

The final unit was about eigenvectors, with a big section about using eigenvectors to recognise the various kinds of non-degenerate conics and quadric surfaces from their equations. It was a bit taxing, but I really enjoyed that section. I find it weirdly satisfying to go through the big drawn-out process of writing the equation in matrix form, diagonalising the matrix, changing coordinate systems, etc, because when you’ve finished all the detective work you get to shout “It’s an elliptic paraboloid!” or whatever it turns out to be. (I promise I won’t actually shout this in the exam.)

Hyperbolic paraboloids by fish2000

I think quadric surfaces are just lovely to look at and think about, quite apart from the fun of identifying them via eigenvectors. I especially like hyperboloids, like the one pictured above, and hyperbolic paraboloids like the ones pictured to the right. Apparently Pringles crisps are also in the shape of hyperbolic paraboloids, so they’re definitely my favourite quadrics! If I was a tutor, I’d make sure to bring several tubes of Pringles to the Linear Algebra day school – I’m sure being bribed with crisps would make even the toughest bit of matrix algebra more pleasant!


Pringles: a legitimate part of any linear algebra study session.

Linear transformations

Unit LA4 has been the most challenging one yet for me. I’ve been working through it on and off for a few weeks now, and it’s been heavy going! Still, the topic itself is very interesting, much more so that the material on linear transformations in MS221.

I think the main reason for that is the fact MS221 concentrates on linear transformations of the plane, so I’m used to thinking of linear transformations as just being rotations, reflections, translations. Now, I love group theory and I love working with symmetries, but M208 has shown me that linear transformations are much more than that.

I mean, we’re all used to linear transformations that map points in \mathbb{R}^2 to other points in \mathbb{R}^2, that’s basically what we got in MS221. And it’s fairly straightforward to imagine mapping points in \mathbb{R}^3, say, to points in \mathbb{R}^2 – like projecting the 2D shadow of a 3D object, or the 3D shadow of a 4D hypercube. But mapping a point in \mathbb{R}^n to a polynomial in P_n? That’s crazy!

And then there’s the wonderful Dimension Theorem:

\textup{Let } t:V \rightarrow W \textup{ be a linear transformation. Then}

\dim \textup{Im}(t) + \dim \ker (t) = \dim V.

Very simple to state, and very useful if you know two out of the three dimensions it refers to. Perhaps I’m easily amused, but I love the fact that if the image of t and the domain of t have the same dimension, then we know that the kernel of t must be the set containing only the zero vector. And if the kernel only contains the zero vector, then we also know that t is one-one! Awesome!

Given how much trouble I had wrapping my head around the various results and strategies in this unit, I was expecting the associated TMA question to be quite gruelling, but it turned out to be fairly manageable. Not easy, but not as fiendish as they could have made it, I think. Or perhaps its fiendishness is subtle and will only reveal itself once I get my marked assignment back…

Anyway, this week I’ll be moving on to Unit LA5, the final linear algebra unit. It’s called Eigenvectors, but I’ve had a peek at the related TMA question, and that seems to be about quadrics – I wonder what the relationship between eigenvectors and classifying quadrics is. Hopefully it won’t take me another month to work though this unit and find out!

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! :)

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!

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