  Let $S$ be an irreducible smooth projective surface defined over an
algebraically closed field $k$. For a positive integer $d$, let ${\rm
Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes
of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\,
\longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed
using the structure sheaf of the universal subscheme of $S\times {\rm
Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$
are isomorphic if the vector bundles ${\mathcal H}(E)$ and ${\mathcal H}(F)$
are isomorphic.
