Large sparse linear systems, like those found in scientific computing, engineering models, and optimization problems, can be solved very efficiently using iterative Krylov subspace methods. The Generalized Minimal Residual (GMRES) method for nonsymmetrical systems and the Conjugate Gradient (CG) method for symmetric positive-definite systems both use orthogonally and efficient subspace projections to make sure that the systems quickly converge while using less memory. Some preconditioning techniques, like partial LU and Cholesky factorization, make them much more useful by making the numbers more stable and speeding up convergence. This paper looks at the mathematical bases, iterative formulas, and preconditioning methods that are necessary for current large-scale computing.