  We derive the $\sigma$-model tachyon $\beta$-function equation of
2-dimensional string theory, in the background of flat space and linear
dilaton, working entirely within the $c=1$ matrix model. The tachyon
$\beta$-function equation is satisfied by a \underbar{nonlocal} and
\underbar{nonlinear} combination of the (massless) scalar field of the matrix
model. We discuss the possibility of describing the `discrete states' as well
as other possible gravitational and higher tensor backgrounds of 2-dimensional
string theory within the $c=1$ matrix model. We also comment on the realization
of the $W$-infinity symmetry of the matrix model in the string theory. The
present work reinforces the viewpoint that a nonlocal (and nonlinear) transform
is required to extract the space-time physics of 2-dimensional string theory
from the $c=1$ matrix model.
