String beta function equations from c=1 matrix model

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.