This work addresses inverse linear optimization, where the goal is to infer the unknown cost vector of a linear program. Specifically, we consider the data-driven setting in which the available data are noisy observations of optimal solutions that correspond to different instances of the linear program. We introduce a new formulation of the problem that, compared with other existing methods, allows the recovery of a less restrictive and generally more appropriate admissible set of cost estimates. It can be shown that this inverse optimization problem yields a finite number of solutions, and we develop an exact two-phase algorithm to determine all such solutions. Moreover, we propose an efficient decomposition algorithm to solve large instances of the problem. The algorithm extends naturally to an online learning environment where it can be used to provide quick updates of the cost estimate as new data become available over time. For the online setting, we further develop an effective adaptive sampling strategy that guides the selection of the next samples. The efficacy of the proposed methods is demonstrated in computational experiments involving two applications: customer preference learning and cost estimation for production planning. The results show significant reductions in computation and sampling efforts. Summary of Contribution: Using optimization to facilitate decision making is at the core of operations research. This work addresses the inverse problem (i.e., inverse optimization), which aims to infer unknown optimization models from decision data. It is, conceptually and computationally, a challenging problem. Here, we propose a new formulation of the data-driven inverse linear optimization problem and develop an efficient decomposition algorithm that can solve problem instances up to a scale that has not been addressed previously. The computational performance is further improved by an online adaptive sampling strategy that substantially reduces the number of required data points.