Construct a linear operator T with minimal polynomial x2(x−1)2 and characteristic polynomial x3(x−1)4. Describe the primary decomposition of the vector space under T and find the projections on the primary components. Find a basis in which the matrix of T is in Jordan form. Also find an explicit direct sum decomposition of the space into T-cyclic subspaces as in Theorem 3 and give the invariant factors.