# 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!

# 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! 🙂