Suppose V is finite-dimensional and T ? L(V). Prove that the following are equivalent.
(a) T is invertible.
(b) $Tv_1, \dots, Tv_n$ is a basis of V for every basis $v_1, \dots, v_n$ of V.
(c) $Tv_1, \dots, Tv_n$ is a basis of V for some basis $v_1, \dots, v_n$ of V.