The Farkas Lemma: Let $A$ be an $mtimes n$ matrix, $binmathcal{R}^m$. Then exactly one of the following two assertions is true:

(1) There exists an $xin mathcal{R}^n$ such that $Ax=b$ and $xge0$.

(2) There exists a $yin mathcal{R}^m$ such that $A^Tyge0$ and $b^Ty<0$.

I want to check which assertion is true for a given $b$. So I constructed the linear programming problem according to the second statement:

begin{equation}

min b^Ty\

s.t. -A^Tyle0

end{equation}

The idea is simple. if the solution $b^Ty$ is great than or equal to $0$, then the first assertion is true; otherwise the second one is true.

But the following matlab implementation always gives me the result $y=0$ or “the problem is unbounded”. Something is wrong, but i have no idea. I would be thankful for any help or references.

```
m = 2;
n = 4;
A = rand(m,n);
b = rand(m,1);
f = b;
A = -A';
y = linprog(f,A,zeros(n,1),(),(),(),());
```