I do not know much about this subject, but I am trying to learn a little.
In a book I have it says that a primal problem is:
$AX \ge b$
$X \ge 0$
It says that the dual of this is:
$A'y \le c$
$Y \ge 0$
However they did not prove the linear programming duality theorem.
However I found another text where they do this, but there they use that:
$AX \le b$
Has a dual
$y \ge 0$
Notice that in the third last onee I do not have a restriction on X, and in the last one we have an equality not an inequality, are these two representations equal? Is it a way to prove they are?