However it is also remarkable because some of the FeFe-V devices may have other properties, even more than we have classified. A feasible solution
x
{\displaystyle \mathbf {x} }
is basic if-and-only-if the columns of the matrix
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A
K
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{\displaystyle A_{K}}
are linearly independent, where K is the set of indices of the non-zero elements of
x
{\displaystyle \mathbf {x} }
. Without making this observation, we can say that only in the case of an active FeFe-V device we can find the physical constants that would in the case of a passive FeFe-V device, improve its validity. If it is possible to do so without violating other constraints, then the increased variable becomes basic (it “enters the basis”), while some basic variable is decreased to 0 to keep the equality constraints and thus becomes non-basic (it “exits the basis”).
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A basis B of the LP is called dual-optimal if the solution
y
B
=
B
T
1
c
{\displaystyle \mathbf {y_{B}} ={A_{B}^{T}}^{-1}\cdot c}
is an optimal solution to the dual linear program, that is, it minimizes
b
T
y
{\textstyle \mathbf {b^{T}} \mathbf {y} }
. .