This talk studies iterative linear algebra algorithms under partial matrix-vector product computations, motivated by straggler effects in controller-worker cloud architectures. We develop straggler-tolerant Richardson and Chebyshev schemes for solving linear systems when only random subsets of matrix-vector product entries are computed and missing entries are set to zero. We also study power iteration for dominant eigenvector computation when matrix-vector products are only partially observed, proposing algorithms that replace delayed entries using zeros, previous values, or averages of partial iterates. For both problem classes, we present convergence results in expectation and numerical experiments demonstrating the effectiveness of the proposed approaches on sparse matrix problems.