For solving linear systems of equations there are several known algorithms. For large linear systems with a sparse matrix iteration algorithms are recommended. But in the case of general n x m matrices the classic iterative algorithms are not applicable with a few exceptions. For example in some cases the Lanczos type algorithms are adequate. The algorithm presented here based is on the minimization of residual solution with subspace decomposition. Therefore this algorithm seems to be applicable for the construction of parallel algorithms. In this paper we describe a sequential version of proposed algorithm first and give its theoretical analysis. The algorithm has some genetic characteristics. After this we will formulate a parallel version of algorithm with subspace decomposition in order to study the details of convergence and the speed up effect of the parallel algorithms. The numerical test results of these algorithms including the speed-up effects of the parallel execution will be shown. The comparison of the classical method and the new approach will also be presented considering the running speed and efficiency too. The computer tests have been carried out with parallel and cluster computing methods as well. © Civil-Comp Press, 2011.