In this note we provide an elementary proof of the following: every automorphismSofL∞is of the formSf=foSzfor allfεL∞. As a first corollary we prove that if there exists a linear isometryTof a linear subspaceAofL∞containingH∞onto such a subspaceB, thenTcan be written as a weighted composition map, namely,Tf= α (foSz) for allfεA, where α εB, |α(λ)| = 1 for all λ in the unit circle andSis an automorphism ofL∞induced byT.As a straightforward consequence, we obtain a description of the linear isometries betweenDouglas algebras.As a second corollary we show that every linear bijectionTofL∞ontoL∞which preserves non-vanishing functions can also be written as a weighted composition map.