Math 4310 Homework

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Homework will be due in class on Mondays, and we will try to return it by the end of the week it's due. Late homework will not be accepted, but the lowest two scores throughout the semester will be dropped. Assignments should be stapled together, with your name clearly written. Working out problems is the way that one really learns mathematics, so if you want to do well in the course I strongly recommend making sure you put enough time into your homework!

Collaboration: First of all, I strongly recommend you at least think about all of the homework problems on your own. You may work with other students to solve the problems, but you must write up your solutions in your own words even if you come up with the solutions collaboratively, and you should write the names of your collaborators on your assignment (this will not affect your grade). Similarly, if you find something in a book, on the internet, or elsewhere that contributes to your solution to a problem, you should again write up the idea in your own words and provide a citation for what you found. Copying solutions from other students or from anywhere else is a violation of the Code of Academic Integrity.

Writing mathematics: Communication of ideas is a central part of doing mathematics, and proofs are best communicated by using language and not just symbols or equations! Accordingly, your homework should be written up in complete sentences with proper spelling, grammar, and punctuation. Some guidelines for writing mathematics well are:

• Your proofs should be a coherent argument that can be read linearly down the page.
• You should specify any variables or symbols you use - for instance many proofs may start out like "Let V be a vector space and v be a vector in V..." but should not start out using the symbols V and v as variables without saying what they are. (Of course you don't need to re-specify symbols used in the problem statement. You also don't need to specify symbols like det and ℝ which represent specific standard things we've used that are not variables).
• Equations or symbolic manipulations you do should be accompanied by an explanation in words of what you're computing and how it contributes to what you're trying to prove.
• You should clearly mention any results from the textbook or lectures you use, and explain how you're using them (e.g. carefully saying what you are applying the theorem to, justifying why the hypotheses are true, and showing how you get from the conclusion of the theorem to the way you are using it).
• To accomplish all of this, you should not be turning in your scratchwork or your "first draft" of solving a problem! You should work out what you need to do to complete the proof of what you're solving, and then write up a solution in a linear manner. Your writeup should look like something that you'd be happy to see in a textbook.

Extended Glossaries: In addition to the "standard" homework problems, many weeks will include an extended glossary. This is an exercise in writing mathematics clearly and formally - you will be given a mathematical term and be asked to define it formally, provide an example and non-example, and prove a theorem involving that term. You should write a page or two that you could imagine seeing as a page in a textbook (consisting of the four things asked for, plus perhaps some extra commentary between them like you'd see in between such things in a textbook). Some examples of glossaries (created by Tara Holm for a previous version of this class) are:

• A very well-written glossary, which is a model to aspire to.
• A less well-written glossary with some comments on how it could be improved.

Resources for writing mathematics in LaTeX: If you've ever wondered how mathematicians type up papers, textbooks, homework assignments, etc., and get them to look so nice, the answer is the LaTeX typesetting markup language. LaTeX is almost universally used by mathematicians, and is also common in other fields where mathematical symbols and equations are used a lot (e.g. computer science and physics). Some students find typing their math homework is a good way to get in the habit of writing well like talked about above. (This is entirely optional, of course!)

If you want to learn LaTeX, there are many resources for doing so. The website latex-project.org has some guides for installing and getting started using LaTeX. The Not So Short Introduction to LaTeX is a nice guide and reference for using it. If you're interested in learning or have questions, you can ask me or one of the TAs.