Matrix/Column rank/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}M}$ denote an ${\displaystyle {}m\times n}$-matrix over ${\displaystyle {}K}$. Then the dimension of the linear subspace of ${\displaystyle {}K^{m}}$, generated by the columns, is called the column rank of the matrix, written

${\displaystyle \operatorname {rk} \,M.}$