Hi, I've proved parts a and parts b, but I'm confused on how to prove part c. I think it should really follow directly from parts a and parts b but I'm lost. In part c, are we assuming that $x_j$ is not a null variable or that $x_j$ is a null variable?
If $x \in P$ and $x_j$ is a null variable then there exists some $p \in R^m$ for which $p'A \geq 0, p'b=0$ and such that the jth component of $p^TA$ is positive. Then $A^Tp>0$ since $(A^Tp)=(p^TA)^{T}$ so we are done.
But what happens if $x \in P$ and $x_j$ is not a null variable? Then there exists some $y \in P$ for which $y_j >0$ then I don't what to do. I'm confused what to do.
I've set up the primal and dual pairs for parts a:
Primal:
max $0^T x$
$Ax=b$
$x \geq 0$
Dual:
min $p^Tb$
$p^T A \geq 0$