A promising approach for speech dereverberation is based on supervised learning, where a deep neural network (DNN) is trained to predict the direct sound from noisy-reverberant speech. This data-driven approach is based on leveraging prior knowledge of clean speech patterns and seldom explicitly exploits the linear-filter structure in reverberation, i.e., that reverberation results from a linear convolution between a room impulse response (RIR) and a dry source signal. In this work, we propose to exploit this linear-filter structure within a deep learning based monaural speech dereverberation framework. The key idea is to first estimate the direct-path signal of the target speaker using a DNN and then identify signals that are decayed and delayed copies of the estimated direct-path signal, as these can be reliably considered as reverberation. They can be either directly removed for dereverberation, or used as extra features for another DNN to perform better dereverberation. To identify the copies, we estimate the underlying filter (or RIR) by efficiently solving a linear regression problem per frequency in the time-frequency domain. We then modify the proposed algorithm for speaker separation in reverberant and noisy-reverberant conditions. State-of-the-art speech dereverberation and speaker separation results are obtained on the REVERB, SMS-WSJ, and WHAMR! datasets.