[[Farkas's Lama]] --- ### **Intro** **Theorem Statement**: > $ > (\exists x > \mathbf 0: Ax = \mathbf 0) \iff > (y^TA \ge 0\implies y^TA = \mathbf 0) > $ The theorem contrasts the relations between a cones and planes in higher dimensions. It has 2 equivalent statement, name the left one L, right one R. * L: $\mathbf 0$ is in the interior of the cone $\{x:Ax = \mathbf 0\}$, which is the cone of all the columns of the matrix $A$. * It has to be that a subset of the columns of $A$ can cancell out with each other, indicating that it's a cone that crosses the origin in a particular subspace. * There is a vector that lies in one side (weakly) for all the columns of $A$, then it has to be the case that it's perpendicular to all columns of $A$. * There has to be a vector $y$ that is perpendicular to the subspace where the cone is spanning.