The implementation of a linear programming interior point solver is described that is based on iterative linear algebra. The linear systems are preconditioned by a basis matrix, which is updated from one interior point iteration to the next to bound the entries in a certain tableau matrix. The update scheme is based on simplex-type pivot operations and is implemented using linear algebra techniques from the revised simplex method. An initial basis is constructed by a crash procedure after a few interior point iterations. The basis at the end of the interior point solve provides the starting basis for a crossover method which recovers a basic solution to the linear program. Results of a computational study on a diverse set of medium to large-scale problems are discussed.