Adapting a method that Freivalds used in the context of bounded-error probabilistic computation, we prove that the languages recognized by log-space unbounded-error probabilistic Turing machines (PL) are log-space reducible to languages recognized by automata of the same type but restricted to use at most log n bits of storage space, for arbitrarily small > 0. Furthermore, we show that the banded-matrix inversion problem Band-Mat-Inv(n ) is log-space complete for PL, for any (0, 1]. This strengthens a result of Jung that Band-Mat-Inv(n) is log-space complete for PL, and may lead to new space-efficient deterministic simulations of space-bounded probabilistic Turing machines.