Let *K* be an arbitrary field and *C*_{n} a relatively free algebra of rank n. In particular, as *C*_{n} we may treat a polynomial algebra *P*_{n}, a free associative algebra *A*_{n}, or an absolutely free algebra *F*_{n}. For the algebras *C*_{n} = *P*_{n}, *A*_{n}, *F*_{n}, it is proved that every finitely generated subgroup *G* of a group *TC*_{n} of triangular automorphisms admits a faithful matrix representation over a field *K*; hence it is residually finite by Mal’tsev’s theorem. For any algebra *C*_{n}, the triangular automorphism group *TC*_{n} is locally soluble, while the unitriangular automorphism group *UC*_{n} is locally nilpotent. Consequently, *UC*_{n} is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup *G* of *UC*_{n} can be arbitrarily large with increasing n or transcendence degree of a field *K* over the prime subfield.