(2) Suppose that V is a finite-dimensional vector space, and let T E L(V).

(a) Show that if Null T Range T = {0}, then V = Null T+ Range T.

(b) Show that if T² = T, then V = Null T Range T.

(c) Give an explicit example of a non-zero operator T € L(R³)such that T2 = T. If you define you operator via a matrix, you should also explain in words what it does to an arbitrary vector.

(d) Show that V = Null T RangeT does not imply T² = T by showing that for any diagonalisable operator S, V = Null S +Range S.