Dimension of $m\times n$ matrices I'm a bit confused on the notion of the dimension of a matrix, say $\mathbb{M}_{mn}$. I know how this applies to vector spaces but can't quite relate it to matrices.
For example take this matrix:
$$ \left[
    \begin{array}{ccc}
      a_{11}&\cdots&a_{1n}\\
      \vdots & \vdots & \vdots \\
      a_{m1}&\cdots&a_{mn}
    \end{array}
\right] $$
Isn't the set of matrices $\mathbb{M}_{mn}$ with exactly one entry $a_{ij}$ set to $1$ on each matrix and $m\times n$ total matrices a basis for $\mathbb{M}_{mn}$? In the sense that we can take some linear combination of them and add them up to create:
$$ a_{11}  \left[
    \begin{array}{cccc}
      1&\cdots&\cdots&0\\
      \vdots & \vdots & \vdots  & \vdots \\
      0&\cdots&\cdots&0
    \end{array}
\right] + a_{12}  \left[
    \begin{array}{cccc}
      0 &1 &\cdots&0\\
      \vdots & \vdots &\vdots & \vdots \\
      0&0&\cdots&0
    \end{array}
\right]\\  + \cdots +
a_{mn} \left[
    \begin{array}{cccc}
      0 &0 &\cdots&0\\
      \vdots & \vdots &\vdots & \vdots \\
      0&0&\cdots&1
    \end{array}
\right]
=  \left[
    \begin{array}{ccc}
      a_{11}&\cdots&a_{1n}\\
      \vdots & \vdots & \vdots \\
      a_{m1}&\cdots&a_{mn}
    \end{array}
\right] $$
So the dimension of all $\mathbb{M}_{mn}$ is $m\times n$?

The term ''dimension'' can be used for a matrix to indicate the number of rows and columns, and in this case we say that a $m\times n$ matrix has ''dimension'' $m\times n$.
But, if we think to  the set of $m\times n$ matrices with entries in a field $K$  as a vector space over $K$, than the matrices with exacly one $1$ entry in different positions and all other entries null, form a basis as find in OP, and the vector space has dimension $m  n$.
