This paper considers the problem of minimizing a nonlinear function subject to linear constraints. The method adopted is the reduced gradient method as described by Murtagh and Saunders, with a conjugate gradient method due to Shanno used for unconstrained minimization on manifolds. It is shown how to preserve past information on search directions when a basis change occurs and when a superbasic variable is dropped. Numerical results show a substantial improvement over the reported results of Murtagh and Saunders when conjugate gradient methods are used.