why do people say "x dimensional vector" when vectors have only one dimension?

Question

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?

Answer 1

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 $d_i > 0$.

Answer 2

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 $\mathbb{R}^n$.

Answer 3

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.