Definition:Rank/Matrix/Definition 3

Definition
Let $K$ be a field.

Let $\mathbf A$ be an $m \times n$ matrix over $K$.

The rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$ is the largest number of elements in a linearly independent set of rows of $\mathbf A$.

Also see

 * Equivalence of Definitions of Rank of Matrix