Avoid Unnecessary Details
Date Written: June 20, 2024; Last Modified: June 20, 2024This one’s easy: just be concise.
This sounds like the most painfully obvious advice there is, but it’s surprising how many people (including myself) fail to abide by it.
To further explain what I mean, there are details that must be included within an exam solution to demonstrate understanding, such as citing a specific theorem and verifying its assumptions. But certain words and phrases do exactly nothing for your solution.
I won’t use real examples, so I made up the following:
- Since \(f:\mathbb{R}^n\to \mathbb{R}^n\) is injective and linear, by rank-nullity we get \(\operatorname{rank} f = n\). Thus \(f\) is an isomorphism.
- Notice that \(f:\mathbb{R}^n\to \mathbb{R}^n\) is injective and linear, and so that means that \(\ker f = \left\lbrace 0 \right\rbrace\), a \(0\)-dimensional subspace of its domain. Heuristically, \(f\) cannot “squash” the domain into a smaller subspace. More preciselly, rank-nullity says that \(\operatorname{rank} f + \operatorname{nullity} f = n\), and since we showed \(\operatorname{nullity}f = 0\), we use our keen intimacy with arithmetic to deduce that \(\operatorname{rank}f=n\). But this is the dimension of the codomain, hence \(f\) is surjective. It’s also injective by assumption, so it’s an isomorphism.
Believe it or not, this is representative of how needlessly wordy and bulky some exam solutions are. Trim off words like “notice that”, “so that means that”, “since we showed”, “our keen intimacy with arithmetic”, etc. Likewise, adding a heuristic explanation (although very valuable for exposition and homework) does nothing for the solution. Writing these out wastes valuable seconds that really add up over the course of an entire exam!
I will say that once, a student wrote “God forgive me” on their exam. This does technically violate the guidelines of this advice, but please do not sacrifice your sense of humour and individuality in pursuit of maximal efficiency on an exam.