I started by making $\displaystyle e_1, e_2,........,e_n$ a basis of U and $\displaystyle f_1,f_2,.....,f_m$ a basis of V.

Transforming the basis of U by T gives $\displaystyle T(e_1),T(e_2),......,T(e_n)$. These vectors may not be linearly independent so sifting them gives:

$\displaystyle T(e_1),T(e_2),......,T(e_r)$ where $\displaystyle r \leq n$. Let $\displaystyle T(e_i)=f_i$ which gives me the basis of V.

Is this the correct way to start? I really can't see another way of doing this question.

EDIT: Sorry for mistyping "Dimension" as the title.