For a matrix, column rank =row rank.
Let A be an m by n matrix. Applying elementary operations on A to get RREF(A)=R, it is noted that row rank r= n- z, where z= dimension of null A.
Let $T \in L(F^n, F^m)$ be defined as Tv= Av. Then, range T= col(A) so rank T=column rank A=c. So n-z=c. This alogwith the earlier equality gives r=c.