Exercise 2.16

Consider the set $S=\left\{x\in {ℝ}^n\mid x_1=\cdots =x_{n-1}=0,0\le x_n\le 1\right\}$. If it is a feasible set of a problem written in standard form, then there exist a matrix $A$ and a vector $b$ such that the zero vector and every vector $\left[\begin{array}{c}\hfill 0,\dots ,0,k\hfill \end{array}\right]$ with $0<k\le 1$ must satisfy the constraints $\mathit{Ax}=b,x\ge 0$.

Since the zero vector satisfies the system $\mathit{Ax}=b$ we get that $b$ must be $0$, in fact $A\cdot 0=0$ for any matrix $A$. Consider the vector $\left[\begin{array}{c}\hfill 0,\dots ,0,2\hfill \end{array}\right]$. It satisfies

where the last equality follows from the fact that $\left[\begin{array}{c}\hfill 0,\dots ,0,1\hfill \end{array}\right]$is feasible. It follows that $\left[\begin{array}{c}\hfill 0,\dots ,0,2\hfill \end{array}\right]$ is feasible too, but this is a contradiction. Therefore $S$ cannot be a feasible set of a problem in standard form.