This lecture covers the Eigen library for linear algebra, focusing on vectors, matrices, arrays, and their sizes, initialization, vector types, dynamic and static sizes, linear algebra operations, solving linear systems, memory management, reshaping, and sparse matrices. It also discusses Eigen::Map for wrapping memory, reshaping, and extracting block matrices. The lecture concludes with arrays, per-component operations, documentation, and the key takeaway message emphasizing Eigen's flexibility and efficiency in C++.