# What is the meaning of Motzkin's theorem?

Theorem: Let $$A$$ and $$C$$ be two matrices. The system of linear inequalities $$Ax<0$$ and $$Cx \leq 0$$ has a solution iff the following equation in $$\lambda$$ and $$\mu$$ does not have a solution$$A^T \lambda + C^T \mu =0$$ where $$\lambda \geq 0$$, $$\lambda \neq 0$$, $$\mu \geq 0$$.

What is the meaning of the $$\lambda \geq 0$$ and $$\lambda \neq 0$$ part? why not write $$\lambda > 0$$?

I am confused! Is there an elementary proof of this theorem?

• $\lambda \geq 0$ and $\lambda \neq 0$ means that each element of the vector $\lambda$ is greater than or equal to 0 and that at least one element of the vector is not 0. This is not the same as saying that all elements of the vector are greater than 0. – Brian Borchers May 19 at 4:45