# 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 known as

The rank of a matrix can also be referred to as its row rank.

Some sources denote the rank of a matrix $\mathbf A$ as:

$\map {\mathrm {rk} } {\mathbf A}$

## Also see

• Results about the rank of a matrix can be found here.