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Extracting inherent patterns from large data using decompositions of data matrix by a sampled subset of exemplars has found many applications in machine learning. We propose a computationally efficient algorithm for adaptive exemplar sampling, called fast exemplar selection (FES). The proposed algorithm can be seen as an efficient variant of the oASIS algorithm [1]. FES iteratively selects incoherent exemplars based on the exemplars that are already sampled. This is done by ensuring that the selected exemplars forms a positive definite Gram matrix which is checked by exploiting its Cholesky factorization in an incremental manner. FES is a deterministic rank revealing algorithm delivering a tighter matrix approximation bound. Further, FES can also be used to exactly represent low rank matrices and signals sampled from a unions of independent subspaces. Experimental results show that FES performs comparable to existing methods for tasks such as matrix approximation, feature selection, outlier detection, and elustering.