Strictly slack dual constraints force zero primal variables ($(A^{T}y{)}_j>c_j⟹x_j=0$)

For the primal linear program maximising $c^{T}x$ with $Ax\le b$, $x\ge 0$, and its dual linear program minimising $b^{T}y$ with $A^{T}y\ge c$, $y\ge 0$, let $x,y$ be optimal. The complementary slackness identity gives $x_j((A^{T}y{)}_j-c_j)=0$. Thus a strict inequality in the $j$th dual constraint forces $x_j=0$. One optimal dual feasible solution can therefore force zero components in every primal optimum. The same argument with roles reversed says that $y_i>0$ forces the $i$th primal constraint to be an equality.