# non-degeneracy in linear programming

How to do the sum?

Consider the standard form polyhedron $P = \{\mathbf{x} | A\mathbf{x}=\mathbf{b}, \mathbf{x} \geq 0\}$. Suppose the matrix $A$, of dimensions $m \times n$,(m<=n) has linearly independent rows, and that all basic feasible solutions are non-degenerate. Let $\mathbf{x}$ be an element of $P$ that has exactly $m$ positive components.

a) Show that $\mathbf{x}$ is a basic feasible solution. b) Show that the result of part (a) is false if the non-degeneracy assumption is removed.

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If this is homework, please use the homework tag. Regardless of whether it's homework, please tell us what thoughts you have about the problem. –  Ben Crowell Feb 5 '12 at 17:11
I find the statement (a) a little confusing. If $\mathbf{x}$ is a vertex of $P$, then $\frac{1}{2}\mathbf{x}$ is also in $P$ with the same support. Am I missing something? –  Louis Feb 6 '12 at 19:33