Definition:Rank (Linear Algebra)

Linear Transformation
Let $\phi$ be a linear transformation from one vector space to another.

If the image of $\phi$ is finite-dimensional, its dimension is called the rank of $\phi$ and is denoted $\rho \left({\phi}\right)$.

Matrix
Let $K$ be a field.

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

Then the rank of $\mathbf A$, denoted $\rho \left({\mathbf A}\right)$, 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 $\mathbf A$.