simplex - algebraic intuition behind the basis of the canoncial form

Question

I'm trying to understand the simplex iteration of a problem with n variables and m technological constraints by following this text. I understand well the geometric interpretation of the iteration - moving between adjacent vertices.

However, I fail to understand the algebraic intuition. Now we're pivoting between adjacent basic feasible solutions = bfs to the standard form of AX + IS = b, X,S >= 0 :

1. Why is it that the bfs must have n variables equal to 0?
2. Why should the rest of the variables form a basis? Isn't a basis a set of linearly independent vectors spanning a sub-space? What are we spanning here, we're just looking for a vertex, aren't we?

Thanks!

Answer 1

1. The BFS should correctly have n-m non-basic variables set to 0. Some of the m basic variables may themselves be 0 but that is a degenerate solution.

2. The basis is indeed the minimal set of linearly independent variables that span vector b. Note that b is an m-vector. So, m of the vectors corresponding to the variables can form a BFS. The total number of variables is n. The number of bases is therefore exponential n Choose m.

Pivoting or moving from one vertex to another is nothing but substituting one of the non-basic variables (the associated column vector) into the basis and removing a pre-existing variable out of the basis. Thus, the basis will always have m (column) vectors.

At any one point in time, given a partition of A into basic and nonbasic variables, such as A = [B|N], then Bx = b and hence the x variables are the coefficients of the span of B that gives b.

It is a fundamental result of linear programming that every vertex of a bounded linearly constrained LP's feasible polyhedron corresponds to basic solution and vice versa. Reference: https://press.princeton.edu/titles/413.html