[Math] Any feasible solution of a linear programming problem can be expressed as the convex combination of Basic Feasible Solutions.

Question

Any feasible solution of a linear programming problem can be expressed as the convex combination of Basic Feasible Solutions of the same.

Is the statement true?

I think the statement is true. Can anyone please help me to understand why this statement is true if it is true?

Answer 1

Consider the following problem.

$\min 0$ subject to $x \in \mathbb{R}$.

It has no basic feasible solution. The statement is false.

Edit:

Consider $\max y$ subject to $ x \ge 0, y \ge 0$.

The only basic feasible solution is the zero vector.

Note that a feasible region can be unbounded but the convex combination of BFS must be bounded. The statement is false unless the linear program has a bounded feasible region.