  Using the natural notion of {\em Hasse--Schmidt derivations on an exterior
algebra}, we relate two classical and seemingly unrelated subjects. The first
is the celebrated Cayley--Hamilton theorem of linear algebra, "{\em each
endomorphism of a finite-dimensional vector space is a root of its own
characteristic polynomial}", and the second concerns the expression of the
bosonic vertex operators occurring in the representation theory of the
(infinite-dimensional) Heinsenberg algebra.
