We start with some necessary concepts in linear algebra: matrices, determinants, eigenvalues, and eigenvectors. We then show how to derive equilibria of mechanical systems. We approximate the Lagrangian about the equilibria and show how to analyze the motion of the mechanical system close to these equilibria. We proceed by showing how to analyze the stability and instability in general cases and find the modes of oscillations in the linearly stable regime. We then apply this general theory to several particular examples, illustrating the cases when there is an instability in the system for a given value of parameters and analyzing the cases when there is a zero frequency mode corresponding to a shift along a particular coordinate.