Theorems of the Alternative for Linear Systems

  • Michael J. Panik
Part of the Theory and Decision Library book series (TDLB, volume 24)


We now turn to an assortment of theorems which are generally characterized as “theorems of the alternative” or, in particular instances, so-called “transposition theorems.” Specifically, a theorem of the alternative involves two mutually exclusive systems of linear inequalities and/or equalities denoted simply as (I), (II). Moreover, it asserts that either system (I) has a solution or system (II) has a solution, but never both. A transposition theorem, which is a special type of theorem of the alternative, asserts for systems (I), (II) the disjoint alternatives of solvability or contradiction given that in one system a particular vector (usually the null vector) is a linear combination of vectors from the other.


Convex Combination Linear Inequality Null Vector Nonnegative Solution Obtuse Angle 
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Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Michael J. Panik
    • 1
  1. 1.Department of EconomicsUniversity of HartfordWest HartfordUSA

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