Assess students' ability to analyze convergence mechanisms and spectral influences for Krylov subspace methods (CG for symmetric positive-definite systems and GMRES for general systems). Items should probe understanding of how eigenvalue distributions, condition number, spectral clustering, Ritz values, and polynomial approximation determine residual decay; effects of preconditioning, restarts, and finite-precision on convergence; and interpretation of numerical experiments for large sparse matrices. Expect moderate analytical reasoning, short derivations, and explanation of strategy for choosing or designing preconditioners.