This chapter is devoted to two related objectives:
the study of certain “rank-preserving” operations on matrices;
the application of these operations and the theory of linear transformations to the solution of systems of linear equations.
As a consequence of objective 1, we obtain a simple method for computing the rank of a linear transformation between finite-dimensional vector spaces by applying these rank-preserving matrix operations to a matrix that represents that transformation.
Solving a system of linear equations is probably the most important application of linear algebra. The familiar method of elimination for solving systems of linear equations, which was discussed in Section 1.4, involves the elimination of variables so that a simpler system can be obtained. The technique by which the variables are eliminated utilizes three types of operations:
interchanging any two equations in the system;
multiplying any equation in the system by a nonzero constant;
adding a multiple of one equation to another.
In Section 3.3, we express a system of linear equations as a single matrix equation. In this representation of the system, the three operations above are the “elementary row operations” for matrices. These operations provide a convenient computational method for determining all solutions to a system of linear equations.