What is the intuition behind Gordan's theorem?

Question

Gordan's theorem:

Exactly one of the following has a solution:

1. $y^TA > 0$ for some $y \in \mathbb R^m$
2. $Ax = 0$ ;$ x \geq 0$ for some non-zero $x \in \mathbb R^n$

I am not looking for the proof. I am looking for a way to wrap my head around the idea/intuition of this result.

Thanks.

Answer 1

Here's one way to look at it.

The first condition can be written as $ A^T y > 0$. Gordan's theorem says that either the range of $ A^T $ intersects the positive orthant, or the null space of $ A $ intersects the nonnegative orthant (at a point other than the origin).

Because the null space of $ A $ and the range of $ A^T$ are orthogonal complements of each other, this result seems geometrically plausible.

Answer 2

This is basically an application of the Hyperplane separation theorem. Take a look at the proof of Gordan's theorem here, where the hyperplane theorem is used.