  We give a linear algebraic and a monadic descriptions of the Hilbert scheme
of points on the affine space of dimension $n$ which naturally extends
Nakajima's representation of the Hilbert scheme of points on the plane. As an
application of our ideas and recent results from the literature on commuting
matrices, we show that the Hilbert scheme of $c$ points on $(\C^3)$ is
irreducible for $c\le 10$.
