Theorems of the Alternative for Linear Systems
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.
KeywordsConvex Combination Linear Inequality Null Vector Nonnegative Solution Obtuse Angle
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