ABSTRACT
In Chapter 4 we described the form taken by the solution of an m × n system of linear equations. The form of this solution depended essentially on the rank of the system, which could be specified in three equivalent ways, as row rank, column rank or inner rank. But since the process of solving the equations also provided the value of the rank, we did not need any special methods for determining the rank. In fact, almost every method for determining the rank of a system is similar (and comparable in length) to the method of solving the equations given in Chapters 2 and 4. Nevertheless, it is often useful to have another characterization of the rank, and in particular to have a criterion for the linear dependence of n vectors in n dimensions. This is provided by the determinant; some preparation is necessary before we can give the general definition, but we shall begin by looking at the simplest cases, n = 2 and 3, where a direct definition is possible.
