Linear Systems of Equations. . . in a Nutshell

Linear mathematical models for equilibrium phenomena yield linear systems of equations. In some cases the model is “lumped” and thus discrete by construction, in other cases the model is continu­ ous but then discrete by approximation. In either case we can ask the same questions: how can we form a system matrix which expresses the mathematical model in the language of linear algebra? how can we characterize the system matrix in terms of structure and mathematical properties? how can we determine if the linear systems of equations has a unique solution — and, if not, identify the cause of non-existence or non-uniqueness? In short, we consider those aspects of linear systems which are necessary precursors to numerical solution. We consider a linear system of n equations in n unknowns: given an n × n matrix A and an n × 1 vector f , we wish to find an n × 1 vector u such that Au = f . In this nutshell we shall address the following topics related to this system of equations: