Linear Algebra: Concepts and Methods

In this short chapter, we aim to extend and consolidate what we have learned so far about systems of equations and matrices, and tie together many of the results of the previous chapters. We will intersperse an overview of the previous two chapters with two new concepts, the rank of a matrix and the range of a matrix.

This chapter will serve as a synthesis of what we have learned so far, in anticipation of a return to these topics later.

4.1 The rank of a matrix

4.1.1 The definition of rank

Any matrix A can be reduced to a matrix in reduced row echelon form by elementary row operations. You just have to follow the algorithm and you will obtain first a row-equivalent matrix which is in row echelon form, and then, continuing with the algorithm, a row-equivalent matrix in reduced row echelon form (see Section 3 .1.2). Another way to say this is:

• Any matrix A is row-equivalent to a matrix in reduced row echelon form.

There are several ways of defining the rank of a matrix, and we shall meet some other (more sophisticated) ways later. All are equivalent. We begin with the following definition:

Definition 4.1 (Rank of a matrix) The rank, rank(A), of a matrix A is the number of non-zero rows in a row echelon matrix obtained from A by elementary row operations.