Is arrow notation for vectors "not mathematically mature"?

Solution 1:

A sign of mathematical maturity is the awareness that truth is invariant under changes of notation.

Solution 2:

A main issue with marking vectors with an arrow is that it is context dependent what is considered as a vector.

Let us decide we mark an element $\mathbb{R}^3$ as a vector, so we write $\vec{v}$ for it. Now, we want to multiply it with a $3\times 3$ matrix, since it is a matrix there is no arrow, or is there? After all the $3\times 3$ matrices form a vector-space and sometimes we use that structure. So, $\vec{A}$?

For example when we show that for $P$ the characteristic polynomial we have $P(\vec{A})= 0$. Wait, haven't we seen the polynomials as an example of an infinite dimensional vector space? Should be put an arrow there, too? Then we can have $\vec{P}(\vec{A})\vec{v}$!

I tried to write this a bit playful. But the serious point is that one really switches the point of view somewhat frequently when doing mathematics, and the notion of 'vector' is not so clear cut, as plenty of structures are (also) vector space.

In somewhat advanced (pure) mathematics, it is thus not very common to use the notation with an arrow to mark elements as vector specifically. But, if in some context it seems useful, there is no problem with it either.

Solution 3:

Like Nox said, it's up to your preference.

Usually, it's fine to not have an arrow over your vectors as long as you define that they are vectors. Although in any case, really, you should define it to be a vector with or without an arrow. Once you say "Let v be a vector" then no arrow is needed. If I remember correctly, one of my linear algebra professors didn't use arrows on theirs while my other professor who is an algebraist uses arrows. If you're using a lot of scalars and vectors, using arrows might be handy. Again, it's a matter of preference, convenience, and the "situation" you're in. If there were numerous scalars and vectors which I was dealing with, I would use arrows so it's easier to spot which is a vector and which is not.

Notation indicates some mathematical maturity but it doesn't say much. I think precision is a greater factor. A "mature" mathematician might put an arrow over v without defining it (though who are we kidding, I doubt such a mathematician exists -- it is mediocre practice). A more mature mathematician would define what they mean by v-arrow (or simply v) at the get-go. So define what you mean and you will be safe.

Solution 4:

Many accurate answers, but I want to take issue with this sentiment:

Ultimately, I just want a yes or no answer, so at least I do not seem like an immature writer when writing my own papers someday.

There is way too much mathematical writing that is obfuscated because the writer wants to seem fashionable. If this issue ever comes up, ask yourself whether your paper is easier to read with or without the notation, and write accordingly. That should be the only question.