I'm working on the following exercise:
Let $a \in \mathbb{R}^{n \times n}$ be a skew symmetric matrix and $ b \in \mathbb{R}^n$ with $c = -b$. Consider the following $(LP)$ named $(P)$
$$\min_{x \in \mathbb{R}^n} c^Tx$$ $$ \text{such that: } Ax \ge b, x \ge 0$$
Show that it is equivalent to it's dual problem $(DP)$
$$\max_{y \in \mathbb{R}^n} y^Tb$$ $$ \text{such that: }y^TA \le c^T, y \ge 0$$
Show that if a feasible solution exists for $(P)$ then there is also an optimal solution for $P$.
I managed to show that both LPs are equivalent by just plugging in the assumptions the assumptions that $A = -A^T$ and $c = -b$ into $(P)$.
But I don' t know how to do the second part. Could you give me a hint?