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A vector like $(7,5,4)$ is said to have the dimension of 3. But I have also heard that the "5" is the "second dimension of the vector". In the later case one would say the vector has three dimensions (plural). What is the correct way of writing/talking?

If the vector has dimension 3 - what do we call the second "direction" (here this would the 5)?

Also, how does one differenciate between the "dimension" and the value? So for example, let's assume the values of the vector describe "length", "height", "age". Then how do I differenciate the sentences

  • "We identified the WORD-A to be length, height and age."
  • "We identified the WORD-B to be 7, 5 and 4."
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    $\begingroup$ What is "dimension of a vector"? Can you give a definition? $\endgroup$
    – cqfd
    May 23, 2019 at 9:47
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    $\begingroup$ We say $v=(a,b,c)$ is a vector in the three-dimensional space $\mathbb R^3$. $a$ is called the first coordinate of $v$, and so on. $\endgroup$
    – user673903
    May 23, 2019 at 9:47
  • $\begingroup$ @Milad: I want to know the usage of the word "dimension" though... $\endgroup$
    – Make42
    May 23, 2019 at 9:51
  • $\begingroup$ @ThomasShelby: Well, I am kind of asking for the definition and the way the words are used. $\endgroup$
    – Make42
    May 23, 2019 at 9:53
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    $\begingroup$ @Make42 As stated by Milad, the use of the word dimension is correct only in the first context. The next has to be the second coordinate is 5. $\endgroup$
    – user376343
    May 23, 2019 at 9:55

2 Answers 2

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A vector does not have dimension. A vector space has dimension. The dimension of a vector space is the number of vectors in a basis (it requires proof that all bases for a given vector space have the same number of vectors (and this also holds for infinite cardinalities)). That is the end of the story with the concept of dimension in this setting: A vector space has dimension; a vector does not have dimension.

For instance, the vector space $\mathbb R ^3$ with the usual structure has dimension $3$. The vector subspace $\{(a,b,c)\mid c = a+b\}$ has dimension $2$. The vector $v=(1,1,2)$ is a vector in each of these vector spaces. The vector itself does not have dimension. It has coordinates. Now, this is where things get crucially interesting. To have coordinates you must first choose a basis. If you view $v$ as a vector in $\mathbb R^3$ and you choose the standard basis, then the coordinates of the vector are $1,1,2$, which just so happen to coincide with the way it is written. But you can choose another basis, and you'll get different coordinates. Further, if you view $v$ as a vector in the smaller subspace, and you choose the basis $\{(1,0,1), (0,1,1)\}$, then now the vector $v$ has only two coordinates:$1,1$.

In short, one should not get confused by the way a vector is presented syntactically. The apparent number of bits of information required to specify a vector typographically says nothing at all about dimension.

Now, the concept of direction is a different story. It's a bit tricky to define direction precisely. A vector is a mathematical entity that within its vector space specifies a direction and magnitude. So, every vector does have a direction, inherently to its existence. If you want to distill just the concept of direction there are several ways of doing so. One, simple, way is to simply divide the vector by its norm, thereby obtaining a canonical direction: a direction with unit magnitude. This requires a norm to exist so it's a bit restrictive. There are other ways that avoid that, but they are all a bit tricky and somewhat unsatisfactory. It turns out to be a lot easier to model mathematically the notion of "direction and magnitude" together rather than just 'direction'.

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  • $\begingroup$ 1. Very helpful explaination. I am comming from a Data Science background. Here, a vector often describes a (more or less) real object, with the coordinates describing the object, having real-life counterpart. I gave an example in the question. Here one might say something like "The dataset has dimension three with the features 'height', 'length', 'age'." Can we transfer this understand/terminology to vectors? Or is this kind of nonsensical? $\endgroup$
    – Make42
    May 23, 2019 at 11:36
  • $\begingroup$ 2. In en.wikipedia.org/wiki/Euclidean_vector#History it is written "The term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s (also called scalar) and a 3-dimensional vector." Is it legit to write "3-dimensional vector"? And later the article writes "In three dimensions, it is further possible to define the cross product, which..." which uses also the plural... This sound different to what you wrote, or did I get something wrong? $\endgroup$
    – Make42
    May 23, 2019 at 11:39
  • $\begingroup$ 3. How would you - terminlogy wise - differenciate between the "third coordinate" and the value in the third coordinate? So for example: Would you say that $(1,2)$ and $(4,5)$ might have "the same coordinates", but "different coordinate values"? Or how would you say this? $\endgroup$
    – Make42
    May 23, 2019 at 11:42
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    $\begingroup$ @Make42 distinguish between the following two notions: The function $x_i: \mathbb R^n \to \mathbb R$ given by $(a_1,...,a_n)\mapsto a_i$ is called the $i$th coordinate function, and $a_i$ which is the $i$th coordinate of the vector $(a_1,...,a_n)$ is just a value of the ($i$th coordinate) function $x_i$. $\endgroup$
    – user673903
    May 23, 2019 at 13:10
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    $\begingroup$ @Make42 the informal meaning of dimension is meant by the Wikipedia page that you’ve mentioned. “In three dimensions” there means in $\mathbb R^3$. $\endgroup$
    – user673903
    May 23, 2019 at 13:17
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I'm gonna try to give an answer to this, but honestly, people do disagree sometimes in terminologies and words they use, and it almost comes down to personal preference at times. The most important thing is not to be ambiguous and to make sure that, even if someone doesn't like the terminology you're using, they'll have no doubts on what it means. People even say stuff that, strictly speaking, are wrong, but they're accepted because there is no ambiguity whatsoever and it's shorter and simpler than the full totally correct term.

Having said that, dimension is, first and foremost, associated to the $\textbf{vector space}$ that you're working on. It's basically how many independent directions you have. These can be $\textit{any}$ independent directions. And although these directions are, as long as they're independent, arbitrary, the $\textit{value}$ of the dimension is not. Strictly speaking, there's not $\textit{the}$ first dimension or $\textit{the}$ second dimension, unless you define which ones those are. And even if you do, I'd prefer to call them first and second directions or coordinates (notice that "direction" and "coordinate" are not the same thing but they imply the same definition here). I guess when you say that "$5$ is the second dimension of the vector", it's pretty clear that you defined $y$ as your "second dimension", but it still sounds terrible, honestly. $5$ is not a dimension, so it's wrong, and there are better ways to say it that don't require an overusage of the term "dimension" for no reason. Like "$5$ is the second component". This is really the only use of "component" in this context that I can think of, so it has a particular well-defined meaning.

On the other hand, you say that your vector $(7,5,4)$ "has $3$-dimensions", even though you're also wrong, strictly speaking, everyone knows that what you're saying really is that it's an element of a $3$-dimensional vector space. But there's no ambiguity whatsoever, so I'd say it's okay. However, personally, I'd say that it's a "$3$-dimensional vector" and that "it has $3$ components", that's how people usually say it, and it does sound better, since you're not saying something that is wrong.

I'm not sure I understood your example. But if I did, then my solution would be to say that elements in your set are of the form (length, height, age), and that you identified a specific one to be $(7,5,4)$.

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  • $\begingroup$ You wrote "there's not the first dimension or the second dimension" - did you mean "there's not the first dimension or the second direction"? $\endgroup$
    – Make42
    May 23, 2019 at 11:03
  • $\begingroup$ I meant that there is no such thing as "the first dimension" in your space. Or "the second dimension". Or "the third dimension", etc. You can fix $3$ independent directions and call them first, second and third, but again, I'd call them "directions", not "dimensions". $\endgroup$
    – Paulo Mourão
    May 23, 2019 at 11:07

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