  A non-self-adjoint operator algebra is said to be residually finite
dimensional (RFD) if it embeds into a product of matrix algebras. We
characterize RFD operator algebras in terms of their matrix state space, and
moreover show that an operator algebra is RFD if and only if every
representation can be approximated by finite dimensional ones in the point weak
operator topology. This is a non-self-adjoint version of a theorem of Exel and
Loring for $C^*$-algebras. Moreover, we construct an example of an operator
algebra for which approximation in the point strong operator topology is not
possible. As a consequence, the maximal $C^*$-algebra generated by this
operator algebra is not RFD. This answers questions of Clou\^atre and Ramsey
and of Clou\^atre and Dor-On.
