The Identities of vector spaces embedded in a linear algebra

In this paper we study the identities of vector spaces embedded in linear algebras. We prove that the identities of the class of all vector spaces embedded in associative algebras do not follow from a finite set of the identities that are true in this class. Similar result is proved for the spaces embedded in Lie algebras. We constructed the example of a four-dimensional algebra over a field of characteristic zero which is a strongly not finitely based. The authors describe strongly nonfinitely based vector spaces that are finite-dimensional associative algebras with unity over a field of characteristic zero.