As a first step let's look at how a vertex may be characterized in algebraic form through the concept of basic solution. We start by considering a linear programming problem written in what is called its standard form. For standard form we mean that all the constraints are equality constraints And all the variables have to be non negative. We also assume that the matrix A has full rank m.
A basic solution is defined starting from a non singular square matrix B of A of dimension m in the following way. Let's partition the columns of A into columns of B and columns of N.
This partition induces the same partition on the variables x of the problem. Now consider a solution obtained by setting all the variables corresponding to columns in N, called non basic variables, to 0. With this choice the variables corresponding to columns in B, called basic variables, are determined in a unique way by the constraints Ax = b and have to be equal to the product of the inverse of B and the r. h s vector b. A solution obtained in this way is called a basic solution. A basic solution is feasible if and only if all its basic variables are non negative. The important point to make here is that a point x is a vertex of P if it is a feasible basic solution.