Systems of Linear Equations

So far we have seen how to find the solution to a system of linear equations AX=B in the case of a Cramer’s system. In this chapter, we consider a general system of m linear equations with n unknowns and we want to find an answer to each of the following three important questions: 1. When does a system of linear equations have a solution? (existence of solutions) 2. If the solution exists, is it unique? 3. How to find a solution? In order to do so, we start by introducing the concepts of the linear combination of vectors, as well as the linear dependence and independence of a set of vectors. Since it is very difficult to verify if a set of vectors is linearly dependent or independent by using the definition itself, it is necessary to develop a mathematical apparatus that can help determine the required relationship among vectors in a simpler way. Therefore, we introduce the term rank and operations of elementary transformations that enable us to convert any matrix into a matrix for which the rank is obvious, without changing its rank. Now that we have the necessary tools, we can present the main results on systems of linear equations, that is find answers to the three questions from the beginning. We start by stating and proofing the important Kronecker-Capelli theorem, which gives the answer to the question about the existence of a solution. We proceed by presenting the Gauss-Jordan elimination method, which enable us to solve any system of linear equations, and we discuss the number of solutions. This method can also be used to find the inverse of a non-singular matrix, which gives an alternative method to finding the inverse. Finally, we present one of the most important applications of the systems of linear equations to economics, which is input-output analysis.