Very confused of dimension of a matrix

Question

I'm very annoyed and confused of the dimension of a matrix. Till now I have thought that the dimension of a matrix is equal to its rank. But this doesn't seem to be the case. Or is that the case sometimes, in special cases..? You see my confusion right here.

Let's take this matrix as example $A= \begin{pmatrix} 1 & 3 & 2\\ 2 & 4 & 4\\ 3 & 5 & 6 \end{pmatrix}$

The rank of this matrix is $2$. I have used Gauss and this was the last result I got (I want keep it short):

$$A= \begin{pmatrix} 6 & 18 & 12\\ 0 & -8 & 0\\ 0 & 0 & 0 \end{pmatrix}$$

But why is the dimension $3$? Really simply because this matrix has got $3$ columns?

Why people say that dimension is equal to rank? Or do they refer to the dimension of an image when they say that?

Please, please I'm very desperate right now to still have no clue about it and I kindly ask for clarification.

Answer 1

The confusion probably comes from the fact that the word "dimension(s)" is used for different things. In the context of vector spaces however, it has a very specific meaning.

A matrix with $m$ rows and $n$ columns is denoted as an $m \times n$-matrix. This gives you the size of the matrix but sometimes "$m$ by $n$" are referred to as the "dimensions" of the matrix. When $m=n$, this number is sometimes simply called the dimension of the square matrix.

There are a number of equivalent ways to describe the rank of a matrix, e.g. the number of linearly independent columns (or rows). In the context of vector spaces, it is the dimension of the column (or row) space of the matrix. The word "dimension" has a very specific meaning in this context, namely the number of elements in a basis for the subspace.

Answer 2

The dimension of a square matrix is simply the number of columns (or rows). The rank of a matrix is the dimension of the vector space spanned by its columns (or rows). So the rank of a matrix is bounded above by the dimension of the matrix. If if is equal, then we say that the matrix is full rank, and then it is invertible.

These definitions still hold for non square matrices. If a matrix has $n$ rows and $p$ columns, then its rank is $\leq\min(n,p)$.

Answer 3

It is better to refer the use of the term dimension only to vector spaces and their elements.

So for a matrix of $m$ rows and $n$ columns we can say that it is a matrix of size $m \times n$ , that represents a linear transformation from a vector space of dimension $n$ (its domain) to a vector space of dimension $m$ (its codomain) and that its rank can be a number $p \le \mbox{min}(m,n)$.

We can also say that such a matrix is an element of a vector space of dimension $q=n \cdot m$, because the set of matrices has a structure of vector space: the vector space of the matrices, that is a different thing from the domain and codomain spaces.