Suppose that T, S :R^n -->R^n are inverses, if { v_1, v_2, ..., v_k} is a basis for a subspace V of R^n and w_1 = T(v_1), w_2 = T(v_2),....,w_k = T(v_k), prove that { 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.