Definition 7

Given a matrix $X$ with $m$ rows and $n$ columns over a field $K$, the dimension of the subspace of $K_d^m$ generated by the $n$ columns of $X$ is called the rank of $X$ with respect to $K$ and denoted by $\operatorname{rg}(X)$.

It can also be said that the rank of $X$ is the maximum number of linearly independent columns of $X$ (as elements of $K_d^m$). Obviously $\operatorname{rg}(X) \leqslant \inf(m, n)$; for every submatrix $Y$ of $X$, $\operatorname{rg}(Y) \leqslant \operatorname{rg}(X)$.

If $E$ and $F$ are two finite-dimensional vector spaces over $K$ and $u$ a linear mapping of $E$ into $F$, the rank of the matrix $M(u)$ with respect to any two bases is equal to the rank of $u$.