An isomorphism theorem for algebras

Suppose we have given two algebras [Símbolo] and [Símbolo] over ą field F and an F- linear  ∅ such that ∅ : [Símbolo]  → [Símbolo] such that ∅  [Formula Matematica] and such that ∅ is a isomorphism of [Símbolo]   and [Símbolo]  as vector spaces over F. We wish to find necessary and sufficient conditions that ∅ be an isomorphism of [Símbolo]   and [Símbolo]   as algebras.