# Dimension and size of an array, matrix, vector

1. For a $1 \times n$ or $n \times 1$ vector, I remember people say it is n-dimensional.
2. For a $n \times m$ matrix, I heard it is said to have size $n \times m$. As to its dimension, quoted from Wikipedia:

a linear map can be represented by a matrix, a 2-dimensional array, and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are zeroth-order tensors.

So is it right that there are two different ways to interpret the dimension of a multidimensional array: one is the dimension of the vector space of the arrays, and the other is ( I don't know how to describe). For example,

• the dimension of a $1 \times n$ or $n \times 1$ vector can be said to be either $n$ or $1$;
• the dimension of a $n \times m$ matrix can be said to be $n * m$ as well as $2$?

If yes, I wonder what cases to use which way and not cause confusion?

Thanks!

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...and here we see that "dimension" is rather overloaded for a word... –  Ｊ. Ｍ. Sep 4 '11 at 14:46
Apropos: one often speaks of a vector as a rank-1 tensor, a matrix as a rank-2 tensor... I wonder what a paper that has to discuss the rank of a tensor and the rank of a matrix does? –  Ｊ. Ｍ. Sep 4 '11 at 14:50

## 1 Answer

I don't find it so bad, though I would agree with J.M. when he noted that the word 'dimension' is a bit overloaded.

In the Wikipedia article, it mentioned that a matrix is a 2D array. The array is 2D - this says nothing about the matrix itself. We are distinguishing between speaking of the dimension of the array and the dimension of the operator, so that's okay.

But there is a different ambiguity that I see. Often, people might say of an $m$ by $n$ matrix that its dimensions are $m \;\text{by}\; n$. As an operator, such a matrix takes an input from a space of dimension n to an output of dimension m. In many courses that I have TAd for, professors have referred to the dimension of a matrix as the rank of the matrix, i.e. the dimension of the image of the transformation. So when they spoke of the Rank Nullity Theorem, they say that the Dimension equals the Dimension of the Transformation added to the Nullity of the Transformation.

So even though I don't think your perceived ambiguity is problematic, I would certainly say that dimension can be used ambiguously. In the end, you ask, how does one prevent this from being a problem (if I may paraphrase you)? Be clear when you use such a phrase. And when reading, look for context. Or ask - somewhere along the way, many mathematicians become capable of interpreting the correct meaning. I suspect this also has the side-effect that mathematicians expect other mathematicians to be similarly capable of performing the magic trick of pulling the correct interpretation out of a hat.

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