Suppose that T, S $\displaystyle :R^n -->R^n $ are inverses, if {$\displaystyle v_1, v_2, ..., v_k$} is a basis for a subspace V of $\displaystyle R^n $and $\displaystyle w_1 = T(v_1), w_2 = T(v_2),....,w_k = T(v_k)$, prove that {$\displaystyle w_1, w_2,....w_k$} is a basis for T(V).
Give an example to show that this need not be true if T does not have an inverse.