Systems of Linear Equations

This chapter presents Gaussian elimination and Gauss-Jordan row reduction that are important techniques for solving linear systems. Attempts to solve systems of linear equations inspired much of the development of linear algebra. The study of linear systems leads to the examination of further properties of matrices including row equivalence, rank, and the row space of a matrix. A linear equation is an equation involving one or more variables in which only the operations of multiplication by real numbers and summing of terms are allowed. When several linear equations involving the same variables are considered together, a system of linear equations is obtained. Many methods are available for finding the complete solution set for a given linear system in which the Gaussian elimination is most important that begins with the augmented matrix for the given system and examines each column in turn from left to right. There are three operations that we are allowed to use on the augmented matrix in the Gaussian elimination method: (1) multiplying a row by a nonzero scalar, (2) adding a scalar multiple of one row to another row, and (3) switching the positions of two rows in the matrix.