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Let $P$ be a polyhedron in $[0,1]^n$ defined by the constraints $Ax \leq b$ for $A \in \mathbb{R}^{m \times n}$, $x \in \mathbb{R}^n$, and $b \in \mathbb{R}^m$.

In the solutions of an exercise, the following is mentioned:

"Since the first $n$ constraints are linearly independent, they correspond to a basic solution of the system which, a priori, may be feasible or infeasible. This solution is obtained by replacing inequalities with equalities and computing the unique solution of this linear system."

So I am quite confused about this:

1) Why does "linear independent constraints" imply that "basic solution"?

2) Is a basic solution not always feasible?

I am thankful for any answer!

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1) Linearly independant constraints define linearly independant hyperplanes at equality ($Ax=b$ for the lines concerned), the intersection of which is a point (or a vector, depending on your way of looking at this).

2) You have no guarantee that this point will satisfy all the other constraints.

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  • $\begingroup$ Great! Thanks a lot! $\endgroup$ – user136457 Mar 16 '17 at 14:10

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