Linear preservers on triangular matrices

Let Tn(F) denote the vector space of all n × n upper triangular matrices over a field F. We characterize those linear transformations T on Tn(F) that satisfy one of the following properties: (1) T preserves rank one matrices; (2) T preserves both singular and nonsingular matrices; (3) T is nonsingular and T(adj A) = adj T(A) for all A Tn(F).