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Novel machine learning algorithms that make the best use of a significantly less amount of data are of great interest. For example, active learning (AL) aims at addressing this problem by iteratively training a model using a small number of labeled data, testing the whole data on the trained model, and then querying the labels of some selected data, which then are used for training a new model. This paper presents a fast and accurate data selection method, in which the selected samples are optimized to span the subspace of all data. We propose a new selection algorithm, referred to as iterative projection and matching (IPM), with linear complexity w.r.t. the number of data, and without any parameters to be tuned. In our algorithm, at each iteration, the maximum information from the structure of the data is captured by one selected sample, and the captured information is neglected in the next iterations by projection on the null-space of previously selected samples. The computational efficiency and the selection accuracy of our proposed algorithm outperform those of the conventional methods. Furthermore, the superiority of the proposed algorithm is shown on active learning for video action recognition dataset on UCF-101; learning using representatives on ImageNet; training a generative adversarial network (GAN) to generate multi-view images from a single-view input on CMU Multi-PIE dataset; and video summarization on UTE Egocentric dataset.