How to Solve Problem Sheets

My fourth semester begins tomorrow. This is first and foremost a reminder for myself. I was foolish last semester. I was not effective. When solving an exercise sheet, there are several opportunities that everybody should take. I did not. I will from now on.

It’s a beginner mistake. I always tell everyone to avoid that. However, I did not follow my own advice. I rushed through the sheets. That’s the worst thing you can do. I was thinking

Let’s finish measure theory in 1 hour! Then I can get to Functional Analysis… and perhaps finish that in 1 hour, too, if I’m quick!

I did not even read through official solutions. I mean, I’m so great and amazing, why should I bother reading solutions and risk learning about new approaches?  (sarcasm alert!)

That is simply not the right way to solve exercise sheets. I did not do my best to understand the problems as well as I can. I only bothered about having a solution as quickly as possible.

The result was ugly. Friends sometimes asked me about certain problems but I would not even remember the exercise. I then usually needed some time to recover my solutions.

Occasionally, there would be an error in the problem itself. For instance, some technical condition was missing that could be overlooked. When that happened, a classmate of mine would frequently ask me about the problem and talk about the error that she recognized. I’m embarrassed to say that not only did I mostly miss the error, I would not even remember the exercise she was telling me about.

That should not happen. Solving a problem someday and not even remembering it in the same week? Not acceptable. Change has to happen. 

1. Generalization

Introduction to Probability Theory, Sheet 1. True or false?

  1. There exists a probability space (\Omega, \mathscr A, \mathbb P) with |\mathscr A| = 8.
  2. There exists a probability space (\Omega, \mathscr A, \mathbb P) with |\mathscr A| = 7.
  3. There exists a probability space (\Omega, \mathscr A, \mathbb P) with |\mathscr A| = 6.

Analysis. One can show that only |\mathscr A| = 8 is possible. Moreso, one may conjecture (perhaps after trying even more examples!) the following postulate:

For finite measurable spaces (\Omega, \mathscr A), the size of the sigma algebra \mathscr A is a power of 2.

Of course, this is much tougher. It’s okay, if one cannot solve it without help. The crucial step is to investigate and pose conjectures. That is indispensable in research later on.

One solution would be to consider (\mathscr A, \triangle) as a \mathbb F_2– vector space, by the way.

2. Investigating Intuition

Functional Analysis, Sheet 5. Let X be a Hilbert space and T \in L(X,X). Prove that the following are equivalent:

  1. \langle Tx, Ty \rangle = \langle x,y \rangle for all x,y \in X.
  2. \|Tx \| = \|x \| for all x \in X.

Analysis. Understanding a result is important. In this case, we have to interpret the equations. The first identity is a result about angles, the second about lengths. Perhaps, the difficult part is really to note that inner products make statements about angles.

Hence, an interpretation would be that in Hilbert spaces the length-preserving operators are exactly the angle-preserving operators.

3. Investigating the Significance of a Result

Measure- and Integration Theory, Sheet 6. A measure \mu on a measurable space X is \sigma-finite if and only if there exists a measurable function f > 0 with \int_X f \ d \mu < \infty.

Analysis. I have seen this characterization in several different sources but not yet witnessed a single application. So the natural question would be: What is it good for other than a decent exercise?

Sometimes, it’s not bad to ask other fellow (more experienced) mathematicians. 

In this case, I was taught that the result could deem as motivation. On \sigma-finite spaces, non-trivial functionals exist – which is a great property to have!

Algebra 1, Sheet 8. Let R = \left\{ \frac{a}{b} \mid a,b \in \mathbb Z, 2 \nmid b \right \}.

  1. Prove that R is a subring of \mathbb Q.
  2. Find the units of R.
  3. Show that R is a principal ideal domain.
  4. Find all prime elements in R.

Analysis. At first glance, this problem is not all too special. It seems like an easy exercise to get used to new terminology. However, I want to stress that even in these cases – or rather especially in these cases – one should try to find more!

It turns out, as a friend studying at Bonn has told me that R is actually “just” some localization. He was surprised that we did not cover that in the lecture, as it’s a key idea that algebraic geometers use all the time.

4. Understanding the Intuition of a Solution

I’m sure, we’ve all experienced this. Sometimes, we “miraculously” solve a problem after having done weird stuff. Why did it work, though? Does it make sense that it actually led to a solution? How could another person have seen that? Or better: how can I see it in two days when a friend asks me?

For me, intuition is the most important part to understand mathematics.

5. Comparing with the Past

Algebra 1, Sheet 14. Prove the following identities for cyclotomic polynomials.

  1. For prime numbers p and r > 0 show \Phi_{p^r}(X) = \Phi_p \left(X^{p^{r-1}} \right).
  2. For odd n \geq 3 show \Phi_{2n}(X) = \Phi_n(-X).

Analysis. This tip is quite simple. Whenever a topic comes up that you have studied in the past, look up your notes from the past! Recall what you were doing then, perhaps you’ll learn something useful from your past self.

Here, I have spent a tiny bit of time with cyclotomic polynomials in high school. So I should listen to my tip and look up past notes.

6. Opportunity for Revision

Numerical Linear Algebra, Sheet 10. For coefficients (\alpha, \beta) \in \mathbb{R}^2 of a regression line y = \alpha x + \beta and the measured data (x_1, y_1), \dots, (x_m, y_m) one can show

\displaystyle \alpha = \frac{\overline{xy} - \overline{x} \overline{y}}{\overline{xx} - \overline{x} \overline{x}}, \quad \beta = \overline{y} - \alpha \overline{x}

where \overline{x} = \frac{1}{m} \sum_{j=1}^m x_j, \ \overline{xy} = \frac{1}{m} \sum_{j=1}^m x_j y_j, etc.

Prove that this formula follows immediately by applying Cramer’s Rule on

\displaystyle \|y-\alpha x - \beta \|_2 = \text{min}.

Analysis. The main culprit here is Cramer’s Rule. A relic for revision. It’s nothing that I have ever studied extensively. So this problem is the ideal call telling us to study Cramer’s Rule!

In general, exercises are great opportunities to finally study topics you have missed out on in the past.

7. Memorization

It’s not to bad to memorize problems or proofs. With that, I don’t mean learning them by heart as you would learn poems in high school.

Instead, from time to time, try recalling the problems that you did this week. Try recalling your solution. Try recalling your additional findings. 

This – I believe – should help making the ideas stick and serves as a further exercise for your brain.

Happy New Semester, everyone!

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