Real Analysis Feedback

As it turns out, that wasn't as bad as I thought it would be. Had some significant time constraints, and some of the concepts of continuity really threw me near the end but it wasn't a total nightmare.

Must admit though, there are still some areas which I feel I still need to "grok". (Hmm... never much liked that word... but replacing it with "understand intuitively" doesn't quite sound right either, but you know what I mean.) Concepts of continuity is one such area that I'll have to spend some extra time on in the revision stage (or maybe even just before my final Analysis section).

In summary, this was certainly the most challenging section yet. Having said that, I did manage to achieve a higher mark than I expected to get in my assignment. As usual, here are some areas in which I screwed up:

It seems intuitive to say that  0 + \infty + 0 = \infty, but this specific rule regarding the sum of these limits was never listed in the set that I can use, so I can't use it, hence I was marked down. I've always enjoyed working with a limited toolset, so this should come naturally after a bit of revision. The main problem is getting that full understanding of the mechanics of each rule so it can become intuitive. In fact, my answer for this particular question (as a consequence) was extremely drawn-out. The proper answer given by my tutor fits on less than half a page of A4. Hopefully more practise will let me see the quick, correct answer more quickly.

I stated Bernoulli's Inequality incorrectly. Absolutely no excuse for that. 🙁

When working out limits of formulas, always state the dominant term before reducing, and always put curly brackets around a sequence (otherwise, it's just a formula).

In some cases I was lazy and claimed that something like \frac{3}{n+2} was a basic null sequence. Although it is quite obviously basic, and quite obviously a null sequence (it converges to 0 as n increases), it's not actually a basic null sequence. So even something as small as this, I need to deconstruct and prove.

Lastly, and this does bear repeating... I need a lot more practise with questions about continuity...

Next up, more group theory!

Time Constraints

I'm coming towards the end of my last section of my Analysis. I'm probably about half-way through the last book. After that, I have to answer the last question in my Analysis assignment, proof read the whole thing and submit it.

I have the whole of today, Monday, Tuesday and Wednesday before I need to submit this Analysis assignment. To make sure I can submit this assignment in time, and get ahead for my next section (Group Theory Part 2), I've taken the Monday, Tuesday and Wednesday as holiday days. That'll make a total of 5 days I've taken off work since the beginning of January just to make sure I'm where I want to be with my study.

There's a ebb and flow to where I am with my study compared to where I should be, and I won't truly understand how well I managed my time until this whole module is over in June, but I sure wasn't expecting I'd have to use holiday days to catch up on study.

One of the most frustrating things about feeling like you're behind is the nagging feeling like you don't have sufficient time to learn the materials sufficiently. This slight panic creeps in and you realise that more than anything else, you need to reach the end of the section (to be able to move on to the related assignment question). So you learn it JUST well enough to move on. Of ALL the sections for this to happen on, it had to happen with Analysis didn't it. The section I was most nervous about.

So in summary, because I've found myself short on time, I'm having to practically rush through the section on Analysis. No time for playing with concepts, no time to study the proofs in depth (or at all in some cases). Just time to get the general gist, and move on.

Something I will say is that I haven't found it necessary to keep a spider diagram of how core concepts relate to one another. Because of the structure of the learning materials I find it fairly easy to see the links and how one theory supports another proof and so on.

What has been a surprise while studying analysis is that there hasn't been a need to write any proofs in quite the same way as I was expecting. It turns out that (so far, at least) there isn't a need to have an in-depth understanding of logical notation or concepts. You need a basic understanding of the "if x, then y" structure, and the consequential converse "If not y, then not x", and so on, but not a great deal more. I'm not sure whether I'm grateful for this or not. Some of the concepts in Analysis require all my brain power, so I'm not sure the additional logical puzzles on top of that would help. Though on the other hand, having a decent foundation in logic feels quite important. I suppose, again, I'll have formed a more solid opinion on this by the end of the module.

Linear Algebra Feedback

Received another good mark for this assignment, which I'm happy about as at the last minute I realised I'd messed one question up so re-did it. Though despite the high mark, I had a lot of great feedback from my tutor.

Overwhelmingly, he mentioned that I had to use more English to explain why I was solving a question in a certain way. This struck a chord with me as the more solutions I read, the less English there appears to be explaining the methods used, and ever since starting this Mathematics journey this is something I've had issue with.  Some people find Mathematics hard to get into because of the certain barriers to entry, ie: the text won't fully explain what a variable is or (case in point) won't explain how to get from one step to the next. Often such a step is taken as "obvious", and I've always considered this as somewhat "elitist". With this assignment I've face-palmingly found myself falling into the same trap.

This is doubly interesting for me, as I actually do really enjoy going into lots of detail in explaining something I've worked on. So in future, I'll be mindful to explain more of my workings in plain English; or at least summarise them in plain English.

Linear Algebra

Blimey. That was a journey... I've always wanted to know more about mathematics in dimensions higher than 3. Turns out I should be careful what I wish for!

Given my background and prior knowledge of 3d transformations, I thought I might find this section rather easy. Though shortly after starting it, I was forced to think about what would constitute an orthonormal basis in four dimensions. Needless to say, this is not only unintuitive, but also impossible to imagine (and as a colleague quite correctly pointed out, impossible to draw!). So instead of relying on intuition, you rely on basic generalised properties of manipulation of bases. After that, you can work in any dimension you want!

Interestingly, amongst all this talk of linear transforms and bases, there was not one mention of cross products. Everything was done without the need for them, which is an eye-opener given the liberal scattering of cross products in my own transformation code!

Perhaps one of the most interesting things I looked at was a description of a basis in four dimensions that you go on to prove is actually representable in three dimensions. Obviously, it's not always possible to do, but this representation of higher dimensions in lower ones was something else I'd been interested in for a while. The 4D window on dimensionality has begun to open!

This section was really fun. It's a shame there is only one section on it this time around. Next up... the dreaded Analysis...

 

A Romance Of Many Dimensions

I decided to buy Flatland: A Romance Of Many Dimensions by Edwin A. Abbott. Glad to say I got exactly what I wanted out of it: a light read!

It mostly takes the form of a light societal allegory, using aspects of multiple dimensions and a little bit of philosophy thrown in there for good measure too. Though I feel it didn't take itself too seriously and did make me smile on occasion. Something this book doesn't do is help you think in four dimensions and higher, though it does make you think a little more about the dimensions we can visualise in our heads!

Entertaining all the way though, it's a great, light read should you be requiring one!

Group Theory Feedback

My second assignment has been marked! Very happy with these results. Though as usual, my tutor has been great and filled my paper with suggestions on how to improve further.

The identity axiom for a group: Often it's really obvious to see that an operation is commutative. Really easy. So easy in fact, that it's often just as easy not to mention that it's commutative. In doing so, you kind of miss out half the answer. Always check! If it works one way, always prove it works the other too!

Another obvious thing that's easy to miss out... mentioning that your result does actually lie within the required working set. ie: if you're working in the \mathbb{Z} universe, you need to explicitly say that your result is also in \mathbb{Z}.

All transformations are relative! I sped through these questions without thinking... silly really. I slipped up here, and never mentioned the point around which something was rotated, or the point around which a reflection line was rotated.

Students apparently screw this up a lot... me included it seems... but answers should be in their correct forms. I was so used to writing Cayley/Group tables as answers, I neglected to realise that the question actually wanted the set which formed the group. Here, effort was spent where it didn't need to be.

Lastly, I need to get better at quickly being able to spot if a Cayley table is Abelian (commutative). This was a silly oversight on my part. Something that's a little less obvious is how to quickly find a group that is isomorphic to my initial (Abelian, in this case) group. I suppose this will come with time and familiarity!

Next up we've got linear algebra. This looks like a big section, so it's good that I'll have the Christmas holidays to break the back of it!

Finally, More Group Theory

I've just come to the end of the second section of study: Group Theory! It was great to get back to this. Since my introduction to it a few modules ago I had been really curious as to what the next steps were with it all, and it's really fun!

Though now I'm back in the same situation as I was when I was introduced to it: I just want to learn more! Group theory will now be shelved until about March I think (like analysis, there are two group theory sections).

Something I noticed though, even group theory at this level is starting to reflect what Alcock was saying in her book about analysis and the building of theorems to prove another further theorem. As such, I may have benefitted from creating that spider diagram I was such a big fan of before. Next up is linear algebra, but after that is the first section of analysis. I'll have to go into that with the view to actually making that spider diagram to get an overview of how all the concepts build on each other.

Feedback - 01

I received the marks back for my first monster assignment! Did quite well as it turns out! But this blog isn't about spouting about my success, it's about the learning process! So here's some of the things I screwed up...

First off, my algebra is clearly rusty as fuck. In one instance put a minus sign in the wrong place AND mysteriously lost a factor of 2 in the progress of my working. In future I really need to re-read my working really carefully (three or four times over it seems), both the hand-written and the full typed-up LaTeX...

Something else I lost marks for was the apparently simple task of graph sketching, either where I hadn't considered asymptotes or had not considered the limits of the domain. Overall I clearly need to be a lot more mindful of whether I'm dealing with \leq or <. When I read those symbols I see them both so often, I frequently gloss over them without properly considering their usage. Again, pretty basic stuff.

With complex numbers I apparently need to be more explicit with my declaration of forms. My polar form was implicit in the answer, but there wasn't anywhere I actually stated it. Silly boy.

I fell down on a proof of symmetry for an equivalence relation. I just wasn't mindful whilst answering this. It is assumed that x-3y=4n. This can be rearranged in terms of y as y=\frac{x}{3}-\frac{4n}{3}. So substituting y, in the symmetrical y-3x results in: 4 \left(-\frac{2x}{3}-\frac{2n}{3}\right). Of course, at this point, proving that what's inside the brackets is an integer is pretty difficult. But that's where I left it. A bit more play would've shown that I could easily have arranged the first equation in terms of x instead which would've resulted in 4\left( -2y-3k\right), which is rather obviously an integer given the initial variables. More exploration required in future...

Lastly, in my last post I mentioned how there was a distinct lack of symbolic existential or universal quantifiers in all this new material. After Velleman, I was so used to seeing them, and working with them appropriately but because they're now not around, I got totally burnt by assuming I had to prove "there exists" instead of "for all" for one question. I suppose I'll be able to get around this with making sure my notes explicitly state whatever quantifier we're actually talking about. Damned English language... Symbols are much more concise! 🙂

Large Intro

Finally submitted my first assignment. It was monstrous. Just over 23 pages of mathematics and sketches of graphs. All of it typed up in LaTeX. Skipping ahead to look at the rest of the assignments, it looks as if this first assignment may very well be the biggest of the whole lot. This is a very good thing as I really don't think I could churn out that much work of a high quality every month.

Glad to say that most of this introduction section I was familiar with. Only really new topic was equivalence relations, which caused some problems initially.

Overall though, what I've found difficult is the apparent lack of logical notation. After reading "How To Prove It" I've become half-decent at making sense of and rearranging logical notation to solve a problem. The difficulty comes in looking at the plain-English description of something in the texts and then having to translate it into logical notation to allow my fussy brain to think about them logically.

Perfect example of this is the definition of a function being "onto". In the text, the definition reads:

"A function f: A \longrightarrow B is onto if f(A)=B".

Which is fine, but the Wikipedia definition reads:

"\forall y \in Y, \exists x \in X such that y = f(x)"

Which for me, gives me a much better idea about how to go about proving if a function is onto. Why leave out the quantifiers? The Wikipedia definition tells me so much more. I suppose translating English into logical notation is just something I'll have to get good at!

Though even after this long intro section, I really feel I need more practise with proofs... I guess this may have to wait until revision time... Next up is the first section on group theory, with an assignment due on November 24th. Onward.