Let $A$ be an $m × n$ matrix. Recall that Gordan’s lemma states that the system $$\{d : Ad < 0\}$$ is inconsistent if and only if the system $$λ ≥ 0 ∈ R ^m , λ \not= 0, A ^T λ = 0$$ is consistent.
Show that the consistency of the second system is equivalent to the statement that $0 ∈ C$ 2. Use a separation argument to prove that $\{d : Ad < 0\} = ∅$ if and only if $ 0 \notin C$, thereby proving Gordan’s lemma
Proof so far
$\Rightarrow$ Let $C$ be the convex hull of the rows of $A$, that is, $C = co(a_1 , . . . , a_m ) ⊆ E$,where $a_i$ is the ith row of $A$. Suppose the second system is consistent. We want to show that $0 \in C$. If $A^{T}=[a_1,a_2,...,a_m]$. Then $A^{T}\lambda=\sum_{i=1}^{m}a_i\lambda_i=0$. Therefore $O \in C$
For the converse direction, if $0 \in C$, then by definition of the convex hull, there exists $\{\lambda_i\}$ such that $\lambda_i \ge 0$ for all $i$, $\sum \lambda_i = 1$, and $\sum \lambda_ia_i = 0$. But then $\lambda = (\lambda_1, ... \lambda_n)$ is a solution the system, and therefore the system must be consistent.
Edit: Whether the statement above is true by definition or theorem depends exactly on how you define "convex hull". But any common definition implies it. For example: convex hull and convex combinations equivalence.