Let $\displaystyle u_{i} '$s be a basis of null T. Then extending it to basis of V gives
$\displaystyle u_{1} ,u_{2} ,...,u_{m} ,\ v_{1} ,v_{2} ,...,v_{n}$
So $\displaystyle Tv_{1} ,\ Tv_{2} ,...,Tv_{m}$ span range T
$\displaystyle \sum c_{i} Tv_{i} =0\Longrightarrow T\sum c_{i} v_{i} =0\Longrightarrow \sum c_{i} v_{i} =\sum d_{i} u_{i} \Longrightarrow $all $\displaystyle c_{i}$'s are zero.
So $\displaystyle Tv_{i}$'s are LI. Hence $\displaystyle n=1$.
So basis of V is $\displaystyle u_{1} ,u_{2} ,...,\ u_{m} ,v$, where $\displaystyle Tv$ spans range T.