Analysis in R^{n}
Autumn 2010
Math 203, Section 30 (InquiryBased Learning)College Fellow: Laurence Field
Recommendations for journal writing
November 3rdOn the first journal I gave some advice on mathematical presentation and style. While there was some improvement in this on the second journal, there are still several points of style which I think your writing should better adhere to.
I have not deducted, and will not deduct, points on these grounds unless they affect the validity of your argument. I will, however, find your work much easier to read if it follows the following guidelines. Good examples are shown in green and bad examples in red.

Make sure everything you write is part of a full sentence starting with a capitalized word (not a mathematical symbol) and ending with a period. You can use mathematical symbols, but not at the start of a sentence.
You don't need to include lots of extra explanation to start each sentence, so long as you use a logically appropriate phrase such as "So", "Let", "Suppose", "Hence", "For", "Therefore", "Because", "We claim that", or similar. This way, the reader knows whether you are claiming that your statement follows as a consequence of the previous statement, is true by assumption, is true for an unrelated reason, or is to be proved later.
It's much better to use "So" or "Therefore" than the implication symbol, ⇒, which can be ambiguous. Does "so a = 0 ⇒ b = 0" mean "so a = 0, and therefore b = 0", or "so
if a = 0, then b = 0"? It could mean either one. It's much better to avoid the ambiguity. 
λv + μw + νv×w = 0 ⇒ <λv + μw + νv×w, v×w> = 0
ν <v×w, v×w> = 0
v×w ≠ 0 ⇒ ν = 0
v×w is linearly independent of v, w 
Let v and w be linearly independent vectors in E^{3}.
Suppose that λv + μw + νv×w = 0 for some λ, μ, ν ∊ R.
Then 0 = <λv + μw + νv×w, v×w> = ν <v×w, v×w>,
since v×w is perpendicular to v and w.
But v×w ≠ 0, as proved earlier, so ν = 0.
Thus λv + μw = 0, so λ = μ = 0.
Therefore v, w, v×w are linearly independent.

λv + μw + νv×w = 0 ⇒ <λv + μw + νv×w, v×w> = 0

Make sure
every variable you use (that is not named in the question) has been introduced with a quantifier such as "for every", "for some" or "there exists". It is usually best to write out these quantifiers in words, although sometimes for brevity it is reasonable to use the symbols ∀ and ∃. You should generally state what set the variables lie in, unless it is completely obvious. 
Don't feel the need to justify every tiny step in your argument, especially if you are using an easy fact that has already been applied repeatedly. Some of the second journals displayed solutions that were, if anything, too long and detailed, perhaps because I expressed dissatisfaction with the explanation in the first journal. Use your judgement and don't put in details that seem excessive; save your time and effort. In particular, there's no need to say "by definition" whenever you use the definition of an object.

w ≠ 0 ⇒ ‖w‖² ≠ 0 by application of the positive definiteness of the norm.
So we are allowed to divide by ‖w‖².
<v  proj_{w}(v), w> = <v  (<v,w>/‖w‖²)w, w> by definition of projection.
<v  (<v,w>/‖w‖²)w, w> = <v,w>  <(<v,w>/‖w‖²)w, w> by bilinearity of the inner product.
<v,w>  <(<v,w>/‖w‖²)w, w> = <v,w>  (<v,w>/‖w‖²)<w,w>, taking out scalars from the inner product.
<v,w>  (<v,w>/‖w‖²)<w,w> = <v,w>  (<v,w>/‖w‖²)‖w‖² by definition of the norm.
<v,w>  (<v,w>/‖w‖²)‖w‖² = <v,w>  <v,w> = 0 by cancellation. 
Let v, w ∊ E^{n} with w ≠ 0.
Now <v  proj_{w}(v), w> = <v  (<v,w>/‖w‖²)w, w> = <v,w>  (<v,w>/‖w‖²)<w,w> = 0.
Thus v  proj_{w}(v) is orthogonal to w.

w ≠ 0 ⇒ ‖w‖² ≠ 0 by application of the positive definiteness of the norm.

Some LaTeX specific hints:
Any time you have an equation that's even moderately complex, it should be a
displayed equation , which you create by putting the equation between \[ and \], rather than dollar signs.When you have a calculation spanning multiple lines, you should align the equals signs (or inequalities) using the align* environment. You will need to use the amsmath package.

We can calculate that
\begin{align*}
a(bccb)+b(caac)+c(abba) &= abcacb+bcabac+cabcba \\
&= 0.
\end{align*}
To comply with the rule that all equations must appear as part of a sentence, you'll need to put the period at the end of a sentence
inside your displayed equation, like for the equation \[ ax = b. \]To get large parentheses and other delimiters in math mode, you can use \left(...\right) where you might have used (...).
People have been handing in scripts with misnumbered exercises, or printing many more pages than needed, because of difficulty in handling LaTeX's autonumbering of theorems. There are two better options: you can remove any definitions or exercises you don't need, and then either

place the command \setcounter{theorem}{32}
just before the theorem/exercise/definition you want to be numbered 33, if this theorem comes after a gap in the numbering; or  change the preamble (or the file "custom.sty", if you're using the template) to replace all the \newtheorems with \newtheorem*s (which suppresses the automatic numbering), and then insert numbers manually by \begin{theorem}[0.0.33] or similar within each environment.
Probably the first option is more straightforward.

We can calculate that