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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?

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    $\begingroup$ $\lambda \geq 0$ 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. $\endgroup$ – Brian Borchers May 19 at 4:45

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