CHAPTER 8 - LINEAR EQUATIONS

Abstract

The variables in a set of linear equations are considered as components of a generalised vector. Elementary row operations are operations that change the entries of one or more rows of a matrix in one or more of many ways, such as replace each entry of a row by k times itself, replace each entry of a row by itself added to k times the corresponding entry of another row, and interchange two rows. A row operation on a matrix M is effected by multiplying M on the left by another matrix, called an elementary matrix. The chapter also provides an overview of the echelon form. An echelon matrix is one in which the first nonzero entry of every row is 1 and in each successive row, this entry of 1 occurs progressively further to the right. The word echelon means rung of a ladder, and echelon matrix is composed of a ladder of 1s, supported by zeros. Every matrix is row equivalent to some echelon matrix. Sets of linear equations are solved by using elementary row operations to reduce an augmented matrix to an echelon matrix.