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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."
Make42
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    What is "dimension of a vector"? Can you give a definition? – cqfd May 23 '19 at 09:47
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    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. –  May 23 '19 at 09:47
  • @Milad: I want to know the usage of the word "dimension" though... – Make42 May 23 '19 at 09:51
  • @ThomasShelby: Well, I am kind of asking for the definition and the way the words are used. – Make42 May 23 '19 at 09:53
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    @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. – user376343 May 23 '19 at 09:55
  • @Make42 You can define the dimension of a vector in $\mathbb R^n$ as $n$, because the ambient space $\mathbb R^n$ is generated by maximally $n$ linearly independent vectors $e_1=(1,0,...,0),e_2=(0,1,0,...,0),...,e_n=(0,...,0,1)$, i.e. every vector $(x_1,...,x_n)$ can be written uniquely as $x_1e_1+...+x_ne_n$. –  May 23 '19 at 09:56
  • @Milad: Ok, and what do we call WORD-A and WORD-B? – Make42 May 23 '19 at 09:57
  • But it’s better not to add extra unnecessary definitions to the literature –  May 23 '19 at 09:57
  • @Milad: I am not sure what you mean by "But it’s better not to add extra unnecessary definitions to the literature" - I cannot imagine that what I am asking for is not already defined and it surely is not unnecessary (afterall I am currently needing it). – Make42 May 23 '19 at 09:59
  • @Make42 what do we call WORD-A? We call it simply WORD-A :D Can you explain what you mean? What’s your book? –  May 23 '19 at 10:02
  • @Milad: You have to read my question: WORD-A is a place holder for the actual word in the example sentence. – Make42 May 23 '19 at 10:12
  • You have identified “WORD-A” with the vector (length, height, age), so we can say that “WORD-A” is a three-dimensional vector with components (or coordinates) length, height, and age. @Make42 –  May 23 '19 at 10:17

2 Answers2

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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'.

Ittay Weiss
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  • 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?
    • – Make42 May 23 '19 at 11:36
  • In https://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?
    • – Make42 May 23 '19 at 11:39
  • 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?
    • – Make42 May 23 '19 at 11:42