Suppose that $\displaystyle T,S:R^n \rightarrow R^n$ are inverses.

If {$\displaystyle v_1 ,v_2 ,..., v_k $} is a basis for a subspace $\displaystyle V$ 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 $\displaystyle T(V)$.

In addition, give an example to show that this need not be true if T does not have an inverse.