Definition:Rank/Matrix/Definition 1

Definition
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 $\mathbf A$.

Also see

 * Equivalence of Definitions of Rank of Matrix