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I am new to linear algebra and have been confused by the terminology "n dimensional vector" that my online course instructor uses. He refers to vectors as an "n dimensional vector" where n is the number of elements in the vector. For example he might say that this is a 5-dimensional vector:

[10 15 20 25 30]

However by definition a vector is 1D. I guess this is an etymology question, but why would people use such confusing terminology? If somebody said something was a "5 dimensional matrix" I assume nobody would think that means it has 5 elements, rather they would think it has 5 dimensions, so why do they talk about vectors differently?

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    $\begingroup$ This is not an etymology question, latter in your course you will learn what is a dimension. $\endgroup$
    – R.W
    Commented Feb 20, 2017 at 3:02
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    $\begingroup$ In symple words a dimension is the number of elements that some vector space has in your base. The base is the set of elements of the vector space that, in some sense, generates all other elements of the space. $\endgroup$
    – R.W
    Commented Feb 20, 2017 at 3:04
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    $\begingroup$ By what definition is a vector 1D? The space spanned by a single vector is certainly one-dimensional, but that’s different from the vector itself. $\endgroup$
    – amd
    Commented Feb 20, 2017 at 4:58
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    $\begingroup$ Thanks everybody for your comments, this is illuminating. As you can see I am very new to this. As for why I called a vector 1D I was just imagining that like if you have a shipping box, that's a 3D object. If you tear off one of its sides, that's a 2D object and maybe you could imagine a matrix on top of it. If you cut off a long thin part from that side, that's a 1D object and maybe you could imagine a vector on top of it. But I am only thinking colloquially and not in math terms. Looks like I have more studying to do. $\endgroup$
    – Stephen
    Commented Feb 20, 2017 at 23:05
  • $\begingroup$ When people talk of the dimension of a vector they're not talking about the shape of the notation (numbers in a row). Unless it's a data structures class and the "vector" is literally just a way of storing a list of numbers. $\endgroup$
    – David K
    Commented Mar 25, 2022 at 5:47

3 Answers 3

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I suppose this is an old question, but as others have stated, the dimensionality of a vector refers to the space of which the vector is a member, in this case $\mathbb{R}^n$.

Though I believe what you are referring to when you say a vector is 1 dimensional is that a vector is a rank 1 tensor. Matrices are, on the other hand, rank 2 tensors, and scalar values are rank 0 tensors.

A tensor of rank $m$ can have dimensions $d_1\times d_2\times\cdots\times d_m$, where $d_i > 0$.

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  • $\begingroup$ Thanks for the response, I think this clarifies things for me. $\endgroup$
    – Stephen
    Commented Oct 2, 2017 at 15:36
  • $\begingroup$ Strictly speaking, in math, a singular value is not a scalar, even it this is sometimes said like this in computer programming. $\endgroup$
    – Make42
    Commented Sep 19, 2021 at 19:00
  • $\begingroup$ A further confusion: The rank of a matrix in linear algebra is defined as the maximal number of linearly independent columns of a matrix. $\endgroup$
    – Youjun Hu
    Commented Mar 10, 2022 at 3:23
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This usually just refers to the dimension of the vector space in which the vector "lives." With that being said, this seems to be nonstandard terminology outside of $\mathbb{R}^n$.

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Dimensionality is a standard terminology in the study of vector spaces, where dimensionality is the number of basis vectors needed to span a vector space. When people say "$n$ dimensional vector", they means that the vector is a member of an $n$ dimensional vector space.

Look basis vectors as basis functions. And coordinates of a vector is the expansion coefficients in terms of the basis functions. When we refer to a vector, besides the abstract symbol (e.g.,A), we use these expansion coefficients, which are called the coordinates of the vector. These coordinates are expressed as 1D array of length $n$ in programming languages. This is the reason why some people call 1D array of length $n$ as an $n$ dimensional vector.

In programming languages, people interpret "dimension" in the same way as you do, i.e., the number of index needed to refer to an element in a container.

In R programming language, a vector has no dimension property and is just a sequence with its elements being of the same type. You can give dimension attribute to a vector to make it become a matrix or more generally a multiple-dimensional array. For example:

> d <- c(1,2,3,4)
> d
[1] 1 2 3 4
> b <- array(d, dim=c(2,2))
> b
     [,1] [,2]
[1,]    1    3
[2,]    2    4

The dimensionality of an array we use in programming languages is actually the rank/order/degree of a tensor. In Fortran, the rank of an array can be checked by function rank().

Also note that rank of a matrix in linear algebra refers to the maximal number of linearly independent columns of a matrix.

It is common that different communities use different words to define the same concept. It is also common that the same word is used in different research areas but referring to different concepts.

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