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In the problem of recovering the 3D structure of a scene from its 2D projections, a fundamental low-level computer vision task is the estimation of the epipolar geometry. Accurate estimation of the epipolar geometry uses computationally expensive iteration schemes based on nonlinear algebraic constraints to deal with the ill-posedness of the problem. Linear techniques are computationally efficient but extremely unstable. Theoretical and practical aspects of linear methods are analyzed and fundamental results are derived from the study. Two main causes of instability are considered. The first one refers to the lack of homogeneity in the input data. To deal with this problem, a highly efficient scaling approach is introduced. The optimality of the technique is proven theoretically and heuristically. It is shown that a second source of instability arises from the linear dependency between rows of the matrix of the linear system. The effect of this problem in the estimation of the essential matrix is analyzed. An additional strategy is introduced to overcome this difficulty. This strategy improves the stability and accuracy of the linear approach even further while reducing the computational cost. Numerical experiments to evaluate the effectiveness of the proposed techniques are reported.