Constrained system of linear equations I have a question that might or might not be trivial for experts from the linear programming world. 
We have a system of linear equations that we want to solve:
$A\cdot x=0$, with the constraint that all variables are non-negative: $x_i \geq 0 ~\forall i$.
The system is underdetermined, i.e. $A$ has more columns than rows, more variables than equations.
Question: 
How do we determine which $x_i$ can never be positive (i.e. are zero) in the solution space. That means, determine those $x_i$ for which
$ x_i = 0$ for any solution.
Thanks!

If the $i^{th}$ column has at least one non-zero component that is of the same sign as every other component in the same row, then $x_i$ must be equal to $0$. This can be iterated, by setting to $0$ any $x_i$ that "passes" the test, removing the corresponding column from $A$, and repeating the test until it "evicts" no more columns.
But this is only a sufficient condition, not necessary (i.e. if it is true, then $x_i$ must be $0$; but $x_i$ could be $0$ for other reasons). Frankly, I do not know if there is a "simple" necessary and sufficient condition.
