This lecture delves into the practical aspects of using the Matlab toolbox Manopt for optimization on manifolds. The instructor demonstrates how Manopt generates random points on manifolds, checks gradient and Hessian computations, and discusses the importance of approximation errors. The lecture also covers the implementation of automatic differentiation for Manopt in Matlab by a student from the course. Various concepts such as gradient checks, approximation errors, and Hessian computations are explored in detail.