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This paper presents a strategy for large-scale SLAM through solving a sequence of linear least squares problems. The algorithm is based on submap joining where submaps are built using any existing SLAM technique. It is demonstrated that if submaps coordinate frames are judiciously selected, the least squares objective function for joining two submaps becomes a quadratic function of the state vector. Therefore, a linear solution to large-scale SLAM that requires joining a number of local submaps either sequentially or in a more efficient Divide and Conquer manner, can be obtained. The proposed Linear SLAM technique is applicable to both feature-based and pose graph SLAM, in two and three dimensions, and does not require any assumption on the character of the covariance matrices or an initial guess of the state vector. Although this algorithm is an approximation to the optimal full nonlinear least squares SLAM, simulations and experiments using publicly available datasets in 2D and 3D show that Linear SLAM produces results that are very close to the best solutions that can be obtained using full nonlinear optimization started from an accurate initial value. The C/C++ and MATLAB source codes for the proposed algorithm are available on OpenSLAM.