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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.

hlin117
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    Arrows on vectors are perfectly fine. Your professor may not have wrote them if there was no need to. For example if I say that $x\in\mathbb{R}^3$, then even though there is no arrow on top of the $x$, you know that it is a vector, since it is an element of $\mathbb{R}^3$. – Sujaan Kunalan Jan 07 '15 at 00:02
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    Most people in mathematics do not write arrow on the vectors, but I do not think that it is really a question of "mature" or "immature", probably just because it is quicker... – Jérémy Blanc Jan 07 '15 at 00:04
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    I know many people think it's "more convenient" or "quicker" to avoid writing arrows over variables when writing math, but how much does half a second of avoiding to write the arrow save you? At the very least, shouldn't people write math for it to be easily comprehended? – hlin117 Jan 07 '15 at 00:07
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    Handwriting: not much, but do that with LaTeX and it will cost you a lot more, especially if you have lots of vectors. – Nox Jan 07 '15 at 00:09
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    @hlin117 I've never even considered using arrows on my vectors, as I never saw it in any text or paper I had encountered. I've written math papers and I didn't even know the LaTeX markup for the vector arrow until I edited someone's question that had it on this site. – Matt Samuel Jan 07 '15 at 00:10
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    Yes, it's pretty rare to see the arrows in published papers. Mostly they just appear in undergraduate textbooks. – Jair Taylor Jan 07 '15 at 00:11
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    @hlin117 I'd guess that it just makes it look busy, and there's usually no problem in determining whether or not something is a vector. There's rarely a chance of confusion, and if there is the author should point it out. – Matt Samuel Jan 07 '15 at 00:16
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    @hlin117: "how much does half a second of avoiding to write the arrow save you?" -- um. It saves you half a second :-) If you're handwriting then it means you're writing two characters instead of one. Since not every character you write is a vector it won't halve your handwriting speed, but it will take something off it and mathematicians speaking/writing in front of a group typically find their own speed of writing somewhat frustrating. Anyway it's unnecessary for the reasons given by others. – Steve Jessop Jan 07 '15 at 10:14
  • Also consider whether vectors are all that difficult to recognise. Should there be a special overhead sigil to indicate when something is a matrix? A function? A random variable? All these things can be used in conjunction with scalars. Or is it sufficient to establish a notational convention to aid readability, typically using certain letters for certain purposes, but ultimately rely on the reader to pay attention when you define things? – Steve Jessop Jan 07 '15 at 10:17
  • @hlin117: Usually there are much bigger difficulties for a reader to navigate than the author's choice to not include arrows. (Less flippantly, and more pratically, if I am reading a paper, and I am having any thought at all about whether a particular object is being treated like a vector or like something else, this is usually an indication that there is a much bigger thing I am not grasping.) – Eric Stucky Jan 07 '15 at 10:28
  • It is baby notation, and using it you can only lose, considering that a vector is by definition an element of a vector space, and usually one considers a lot of vector spaces. It is like the newbies notation $1_G$ for the identity element $1$ of a group $G$ – Mister Benjamin Dover Jan 07 '15 at 19:59
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    In my hand writing I always put arrows and in my LaTeX I always bold, even for vector-valued functions. To me it's like putting a capital R in mathbb font or giving it an extra vertical line. Without it, it doesn't denote the real numbers. With it, a mere glance makes everything clear. – Todd Wilcox Jan 07 '15 at 22:17
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    Since nobody seems to have mentioned it, if you want to know how to notate something then read books until you find one you like, then copy that. – Sean D Jan 08 '15 at 13:40
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    @Matt Samuel: The use of arrows is pretty universal (on blackboards) in physics and engineering, at least at the undergraduate level. In fact, I believe arrows were even used in graduate quantum mechanics and relativity classes I took (back in 1980). – Dave L. Renfro Jan 08 '15 at 21:44
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    After thinking about this some more, it occurs to me that my upper level and graduate level physics classes probably used underlines or overlines for vector valued quantities, rather than arrows. But they were written in a different manner than real (or complex) valued constants, variables, and functions. – Dave L. Renfro Jan 08 '15 at 21:54
  • @hlin117 Hm. Is this in reference to Boyd's class? – Guillermo Angeris Jan 10 '15 at 07:16
  • @GuillermoAngeris Are you from UIUC as well? I'd rather not state my professors in public, but you could message me privately if you're interested. – hlin117 Jan 10 '15 at 16:17
  • One note I want to add since I haven't seen this mentioned explicitly: it's awfully difficult to handwrite in boldface. It's not necessarily impossible, but in blackboard situations when you want to avoid confusion over whether something is a scalar or a vector (you'll see this a lot in early linear algebra courses) an arrow as an explicit marking of vector-ness can be immensely useful and it's much easier to recognize than any effort at boldface. – Steven Stadnicki Jan 11 '15 at 02:03
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    For what it's worth: in physics it's only customary to put arrows above 2D or 3D vectors, but not anything else: not elements of any generic vector space, not 4D space-time vectors, not quantum-mechanical state vectors, etc. – Szabolcs Jan 11 '15 at 04:02
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    I'd like to mention that it's probably more common in your machine learning class because students are not expected to have a mathematical background (i.e. they might have a programming background). The arrow notation makes it clearer for those who would not know that x∈R3 is a vector because they don't normally use mathematical notation. – Pharap Jun 18 '15 at 21:33
  • I think the put-down subtext (intended or not, easily encountered in the wider environment) should be at least (no pun intended) acknowledged here. I think everyone knows what I mean: Q: What is a vector? A: Something with an arrow over it! (Tee-hee-hee, those silly mathematically backwater physicists!) –  Dec 12 '18 at 08:35

18 Answers18

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

Rob Arthan
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    Kudos, for the most correct comment here. – hlin117 Jan 07 '15 at 00:34
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    But that doesn't mean that notation is irrelevant. On the contrary, a huge part of mathematical progress consists of finding notation that will support efficient communication of mathematical ideas to people. Discarding all that just because different notations are formally equivalent amounts to doing everyone a disservice. – hmakholm left over Monica Jan 07 '15 at 17:40
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    I couldn't agree more. A good choice of notation is very important. (Personally, I like bold for vectors.) – Rob Arthan Jan 07 '15 at 17:51
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    While this answer is true, I don't think it's an answer to the question asked. – LarsH Jan 07 '15 at 22:06
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    The answer was intended as an encouragement to the OP not to be timid about his or her choice of notation. – Rob Arthan Jan 07 '15 at 22:34
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    @RobArthan, I do too, but I'm terrible at writing bold ;) – Paul Draper Jan 08 '15 at 05:04
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    @PaulDraper: I preferred wavy underline instead of an overhead arrow anyway. It's the oldschool manuscript indicator for bold text, so in that sense it's actually very easy to write bold. – Steve Jessop Jan 09 '15 at 04:38
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    Truer words never spoken. – nakajuice Jun 15 '15 at 17:52
  • I remember there was also a similar slogan for differential geometry :) – freishahiri Jul 22 '15 at 22:20
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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.

quid
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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.

August
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  • Excellent point, about the discussion of "precision". – hlin117 Jan 07 '15 at 00:27
  • This is a great answer, but I would replace the word constant with the word scalar. You can have constant scalars and vectors as well as variables that are scalars and vectors. In fact, this entire discussion is about variables. – clahey Jan 09 '15 at 18:13
  • I realized that but then didn't change it. I'll do that right now. Thanks for the feedback! – August Jan 10 '15 at 05:09
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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.

hunter
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    Maybe not the only question, but I agree that clarity is much more important than projecting a professional image. – LarsH Jan 07 '15 at 22:11
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    But to make a paper easy to read you should often use one standard notation. Still, I agree that the "trying to sound fancy" is the wrong concern. – Blaisorblade Jan 08 '15 at 00:15
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No. (These are filler characters to reach the 30 limit.)

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    -1: why no? you need to explain – user153330 Jan 07 '15 at 13:52
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    You didn't read the question carefully: "Ultimately, I just want a yes or no answer". I seem to be the only one who did. -1 to you for your sense of humour. –  Jan 07 '15 at 15:00
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    I have to say, I enjoy @Yves Daoust's taste in humor. – hlin117 Jan 07 '15 at 15:35
  • @YvesDaoust you're not only answering the hlin117 but also all of those who may get into this question will be very unsatisfied to read such answers who add no value – user153330 Jan 07 '15 at 18:59
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    That just makes it a poorly worded or poor question. User may as well have received a "yes" and "no", what then? You'll notice the upvotes in favor of "no" garnered around the best explained answers, as they would on any healthy SE site. – djechlin Jan 07 '15 at 19:20
  • I’d like to add that a quantal or yes/no answer isn’t necessarily impoverished or lacking substantiation (as this one is), just definitive. In fact, the inherent assumption on SE is that everyone should validate their posts with background information. I would go so far as to say that if we are going to spend our time and energy spreading knowledge then we should dispel the belief that end-all-be-alls even exist. I had originally upvoted this answer, but I think I’ve changed my mind. – gen-ℤ ready to perish Jan 16 '18 at 15:39
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    Anyways, ${}{}{}{}{}{}{}{}{}{}{}{}{}$ is a great way to reach the character limit – gen-ℤ ready to perish Jan 16 '18 at 15:42
  • @ChaseRyanTaylor: nice trick, thanks ! –  Jan 16 '18 at 16:06
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    I use 'Hip hip hooray for Esperanto!' as the filler characters to reach the required limit. –  Dec 12 '18 at 08:30
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    @EulerSpoiler: cool. How does that translate to Esperanto ? –  Dec 12 '18 at 08:56
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    @YvesDaoust: Vivu Esperanto! –  Dec 12 '18 at 09:57
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    @EulerSpoiler: what a concise language ! –  Dec 12 '18 at 10:00
  • @YvesDaoust: Actually, on average Esperanto is a bit wordier than English, because it is typically used in an international context, where culture-specific conventions cannot be employed for the sake of convenient brevity. –  Dec 12 '18 at 16:14
  • @EulerSpoiler: Fakte, averaĝa Esperanto estas iom pli vorta ol la angla, ĉar ĝi estas kutime uzata en internacia kunteksto, kie kulturaj specifa konvencioj ne povas esti uzataj pro oportuna breveco. –  Dec 12 '18 at 18:47
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    @YvesDaoust: Efektive, averaĝe Esperanto estas iomete pli multvorta ol la angla, ĉar ĝi estas kutime uzata en internacia kunteksto, kie kulture-specifaj konvencioj ne povas esti uzataj por oportuna mallongeco. –  Dec 12 '18 at 21:33
27

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.

Chris Godsil
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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".

Markus Shepherd
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    I really can't agree with this, since you seem to be implicitly asserting that the arrow notation can only be used to represent an element of $\mathbb{R}^n$ (perhaps even specifically $\mathbb{R}^3$). As far as I know, that's not the case. – David Z Jan 07 '15 at 20:54
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    Obviously not, you are free to use any notation that pleases you. Paint a heart over your vectors if you like. Our a smiley face. Nobody can restrict your creativity. But would you be clearer to your readers? I doubt it. The arrow convention comes from just that: you can visualise vectors in $\mathbb{R}^n$ (or $K^n$ if you prefer) as arrows in a coordinate system. I doubt you find any author using arrows for functions, matrices, or any vectors other than $K^n$. – Markus Shepherd Jan 07 '15 at 21:11
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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.

Steve Jessop
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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.

Alexander Gruber
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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.

user7530
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  • How does the arrow conflict with the hat here: $\hat{\vec p}$? – Ruslan Jan 10 '15 at 16:10
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    Sure you can stack multiple markings over the variable. It looks super-busy and unprofessional (moreso than just the arrow alone). – user7530 Jan 10 '15 at 20:55
  • As physicist, I agree. But I will not relate it with maturity. The arrow implies a lot more than what linear algebra is about. In particular an arrow implies that you have a metric, and you are not going to distinguish between afine and non-affine vectors. So, I agree that mathematicians shouldn't use the arrow, specially when teaching linear algebra. they may use it when teaching other things, like geometry or differential geometry. – alfC Mar 24 '16 at 22:59
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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).

Nox
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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.

Tim Raczkowski
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  • To follow up with @Jair Taylor's comment above, then why don't published papers use arrows, and mostly undergraduate textbooks? – hlin117 Jan 07 '15 at 00:15
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    Because in typeset mathematics vectors are usually indicated by bold characters, and scalars by normal characters. When hand-writing mathematics, you can't do bold characters, so you use an arrow to differentiate from scalars. – David Jan 07 '15 at 01:02
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    It is by no means standard in published papers that vectors are bold-face, etc. – paul garrett Jan 08 '15 at 21:56
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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.

jinawee
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    I would find this convention horribly confusing. – user7530 Jan 08 '15 at 21:15
  • I don't think anyone uses this convention, certainly not in contemporary published papers. – paul garrett Jan 08 '15 at 21:57
  • @paulgarrett although there may be no one using exactly arrow notation with modulus without arrow, such distinction is still used when vectors are bold. E.g. in Landau&Lifshitz books (admittedly not mathematics) $\mathbf{v}$ is a vector while in the same chapter $v\equiv|\mathbf{v}|$ without special notice. – Ruslan Jan 10 '15 at 16:15
  • @Ruslan, I cannot speak for contemporary physics styles, but this convention seems archaic and text-book-y, reminiscent of styles from several decades past. Not that it's a bad thing, but that contemporary styles outside of textbooks may not follow that, if, for example, the questioner is wondering about contemporary stylistic tendencies. – paul garrett Jan 10 '15 at 16:27
  • @paulgarrett I was considering the textbook context, like Feynman's lectures:http://www.feynmanlectures.caltech.edu/I_11.html. Btw, I've seen boldface to denote three dimensional vectors in space, and normal font for four dimensional vectors (in space-time). – jinawee Jan 10 '15 at 16:41
  • Who uses ISO standards for papers? – anomaly May 19 '15 at 17:50
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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.

Keith
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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}

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

Ooker
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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.

user141592
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    Boldface for vectors is also a textbook convention, nowadays... in part, again, because nearly anything can be construed as a vector, so the distinction(s) have faded. – paul garrett Jan 08 '15 at 22:00
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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.

Loki Clock
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