Is arrow notation for vectors "not mathematically mature"?

Question

Assuming that we can't bold our variables (say, we're writing math as opposed to typing it), is it "not mathematically mature" to put an arrow over a vector?

I ask this because in my linear algebra class, my professor never used arrow notation, so sometimes it wasn't obvious between distinguishing a scalar and a vector. (Granted, he did reserve $u$, $v$, and $w$ to mean vectors.) At the same time, my machine learning class used arrows to denote vectors, but I know some other machine learning literature chooses not to put arrows on top of their vectors.

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.

Answer 1

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

Answer 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.

Answer 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.

Answer 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.

Answer 5

No. (These are filler characters to reach the 30 limit.)

Answer 6

A $2\times2$ real matrix is a vector in the vector space of $2\times2$ matrices. The polynomial $t^2+t+3$ is a vector in the vector space of polynomials of degree at most three. The function $\sin(x)$ is a vector in the space of continuous functions. Should we put arrows on these?

The problem is that almost everything is a vector.

Answer 7

I think the confusion lies in the term "vector". Most people think first of an element in $\mathbb{R}^n$ when they hear vector - maybe they were introduced to vectors like that in school, or they are just the most common they work with. And then the arrow above the symbol makes perfect sense.

But in mathematics, a vector is defined as an element in a vector space, and that can be pretty much any structure. $\mathbb{R}^n$ is just the most common example, but functions, matrices, every field (so in particular scalars), and many more are vectors as well.

So as far as I'm concerned, if you say "let $v$ be a vector", an arrow would be misleading, if not wrong, since it implies the wrong image: by no means $v$ has to be an "arrow".

Answer 8

I think it's a maturity issue in one sense, which is that when you're first introduced to vectors is likely to be the first time you're exposed to the fact that there are two different operations, both called multiplication and both written the same way. You need to learn that you can multiply scalar by scalar, or vector by scalar, but not vector by vector. And perferably understand why. Same issue for addition, you can add two scalars or two vectors, but not a vector to a scalar.

Therefore, there's a critical period where you will get very confused if you don't carefully distinguish between them. Consistently writing vectors and scalars differently will likely help with this.

However, once you understand the difference and have become accustomed to distinguishing them in your mind, it's less useful to write them very differently, and might become an annoying overhead. Reserving letters still helps, and usually is enough to remember what everything is. So, putting arrows or underlines everywhere feels like too much. In typeset text you might continue to use bold for vectors because it doesn't make things look much more busy on the page, but in writing I think most people drop it.

I don't think this means the ones who don't drop it are immature, though, just that those who are immature probably shouldn't drop it yet.

Furthermore, like quid's answer says, you will eventually need to deal with more than one vector space at once, at which point having a notational convention that distinguishes "vectors in my vector space" from "everything else" is not sufficient anyway. So as you mature you have to rely on other means to understand what type of entity everything is.

Perhaps off-topic, but there's a related thing in computer programming called "Systems Hungarian notation", where every variable name contains a prefix denoting the data type of the variable. Almost everyone hates this, because in practice it's too much repetition of the same information. It's actually quite difficult to strike the "right" balance because it's a matter of taste. Using mathematical block capitals for special entities $\mathbb{R}$, latin vs. greek letters for vectors vs. scalars, $n$ for an integer, $q$ for a rational, $x$ for a real, $v$ for a vector, $a$ for a coeffient and $f$ for a function, uppercase for matrices, all seem perfectly sensible to most mathematicians although none is essential. Hanging decorations off your letters, though, starts to cross a line and so fewer people bother.

Answer 9

I will buck the consensus here and say that yes, all else being equal, putting arrows on vector quantities is a slight signal that either the author or the intended audience are beginning students or physicists.

There are several reasons more "mature" math authors avoid the notation:

• Putting an arrow over everything when writing Tex becomes annoying very quickly;
• If the paper is well-written, there should be no confusion about what is a vector or scalar quantity, regardless of the presence of the arrow;
• The space above variable names is valuable real estate, and the arrow conflicts with hats, tildes, etc. that you may want to use for some variables instead.

But again, it is only a slight signal. Use whatever notation you feel is best to ensure that your writing can be easily understood by your audience.

Answer 10

In my experience, $\overset{\rightharpoonup}{v}$ and $\mathbf{v}$ are used more often in science and engineering (even at the research level). This explains why they are often used in lower division mathematics courses, since at this point most of the class is not a mathematics major. As you proceed through mathematics, you begin to see vector spaces like real numbers and function spaces, and it makes less sense to think of vectors as "different" from the other things you're working with. Once you start hearing the word "module" enough, the notation has almost completely disappeared. The only exceptions I have seen are in differential geometry and applied math. I don't think the notation is so much mathematically immature as it is associated with "spatial" thinking.

Answer 11

In my experience this question is really just that: notation.

Pick one and stick to it (at least throughout each publication). Most of my aquaintances who are mathematicians just skip the arrows, as well as subscripts (i.e., $L^2$ in $||\cdot||_{L^2}$) with an explicit remark that it is obvious.

I prefer to keep the arrows for reasons of clarity but it can be a pain to keep writing them if you have lots of vectors (and only vectors).

Answer 12

In a linear algebra class, people tend to omit the arrows, because the notation becomes cumbersome otherwise, but there is nothing wrong with using them.

Answer 13

A possible advantage of the arrow notation to denote vectors in $\mathbb{R} ^3$ is that if you are using the vector $\vec{v}$, then its modulus can be written as $v$, instead of $\vert v\vert $.

Anyway, the ISO 31-11 accepts both $\mathbf{v}$ and $\vec{v}$. The APS recommends boldface only.

Answer 14

It seems to me that the arrow notation is used in undergraduate (and dare I say Engineering?) textbooks to emphasize in the notation that the variable is not simply a scalar.

@august correctly stated that once we say "Let v be a vector" it is redundant to have the notation provide a reminder.

However, mathematicians used the phrase "Let x be a Foo" for a huge variety of values of "Foo", and these "Foo" become increasingly abstract. This compares with elementary mathematics where we are taught in terms of mathematical objects we can relate to the tangible physical world.

So it seems to me that use or otherwise of the arrow notation might be a natural side effect of "mathematical maturity" - so what? There is nothing wrong with it.

I'd suggest that in determining whether to use it or not, you should be thinking in terms of your readership's understanding of what you have written, not any personal impression they might draw. If the arrow notation will make things clearer to them use it.

Answer 15

At least in Physics, vectors are designated as such via arrows (or underlines and matrices underlined twice) very often since you also regularly need their absolute values, which are then simply written as the same variable without decorators, e.g.

\begin{align} \underline r &= (x, y, z), \\ r &= |\underline r| = \sqrt{x^2 + y^2 + z^2} \end{align}

Answer 16

One way to avoid this is using Greek letters for scalars.

Answer 17

I'd say that the key thing when writing mathematics is that your notation is clear. In books and papers, the standard notation (in the books I've read at least) is to use bold for vectors and normal letters for scalars, but when writing on the board we can't really do bold characters, so various lecturers use different styles to indicate vectors. Which one you use is up to you (I have seen arrows, underlines and overlines used), as long as you are consistent and clearly explain you notation when you start. For typed math, I said the standard is bold. This doesn't mean that arrows are wrong, but they are more common in texts aimed at engineers or physicists where vectors generally mean something with a magnitude and direction. A good sign of mathematical maturity in my opinion is clarity. As long as you are clear about your notation, you will be fine whichever one you use.

Answer 18

There's a reason I avoid it, which is that I hand write almost all my mathematics, writing in bulk, and it slows me down to write the arrows overhead. It helps me get down detailed thoughts quickly to use the same notation for writing those notes that I would use when I'm writing a full proof or reasoning from a prepared vantage point. Boldface isn't better, at least for speed (Maybe clarity? I have scribbly handwriting.), but knowing some calligraphy helps especially to cheat and have distinctions clearly hold within the text. Either way, if I don't have two sets of things using the same letters, I'll just use unadorned, pen-weight letters, and just draw them kind of formally clean. I have a feeling I'm not alone in this, and that the adoption of this practice would influence those exposed to its effect on an author's style, which would be another explanation for the large amount of authors there are that don't use arrows or boldface to distinguish vectors from other things. It's also normal if someone doesn't do that just because they don't feel the need to distinguish vectors' names from scalars' names, or have chosen some other notation sufficient to distinguish vectors and scalars in a formula, like a dedicated symbol and order of arguments for the scalar multiplication operator (useful for talking about rings as a module over themselves). The hat over unit vectors or just normalizations of other vectors, on the other hand, is useful for keeping your number of symbols small and showing by the name when things are related.