\begin{definition}[Definition:Rank/Matrix/Definition 1]
Let $K$ be a field.
Let $\mathbf A$ be an $m \times n$ matrix over $K$.
Then the '''rank''' of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$.
That is, it is the dimension of the column space of $\mathbf A$.
\end{definition}