Adaptive matrix completion has been at the center of compressed sensing research for a long time. Adaptivity interacts with the problem both in multi-phase and single-phase settings. One of the modern ideas in adaptive matrix completion is related to sparsity-number. It has shown that the sparsest vector of column and row space carries crucial information regarding low-rank exact completion. However, currently, the proposed method requires many phases going over columns of the matrix. Therefore, in this paper, we suggest an algorithm that studies how to recover the column space in a single phase using the idea of sparsest vectors. Moreover, we extend our method to make it robust to sparse noise in some columns. We provide experimental evidence to illustrate the performance of our algorithm.