ABSTRACT
The goal is to find a particular object, chosen from among a large collection of contenders, which satisfies the governing constraints of the problem. The collection of contending objects is often a vector space, and although individual elements of the set can be called by many different names, it is typical to call them vectors. In a vector space, there are many different choices for a basis. The main message is that by a careful choice of basis, many problems can be “diagonalized.” This is the main idea behind transform theory. The natural frequencies of vibration of a lattice are also called the resonant frequencies because, if the system is forced with a forcing function vibrating at that frequency, the system will resonate, that is, experience large amplitude oscillations which grow as time proceeds. This observation is the main idea behind a variety of techniques used in chemistry to identify the structure of molecules in liquids, solids, and gases.
