  We explore consequences of $W$-infinity symmetry in the fermionic field
theory of the $c=1$ matrix model. We derive exact Ward identities relating
correlation functions of the bilocal operator. These identities can be
expressed as equations satisfied by the effective action of a {\it three}
dimensional theory and contain non-perturbative information about the model. We
use these identities to calculate the two point function of the bilocal
operator in the double scaling limit. We extract the operator whose two point
correlator has a {\it single} pole at an (imaginary) integer value of the
energy. We then rewrite the \winf~ charges in terms of operators in the matrix
model and use this derive constraints satisfied by the partition function of
the matrix model with a general time dependent potential.
