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User:Ben Spinozoan/Wronskian&Independence

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Proof

[edit]
  • In the language of first-order logic, the set of functions is linearly independent, over the interval in , iff:
, where .
Expressed in disjunctive normal form (DNF) the above definition reads:
,
where is shorthand for the statement occurring immediately before the iff (note that negation of gives the correct statement for linear dependence).
  • The text of our theorem "If the Wronskian is non-zero at some point in an interval, then the functions are linearly independent on the interval", now translates as
,
or, in DNF,
,
where is the value of the Wronskian at the point .
  • The following statement summarizes the situation when Cramer's rule is applied to the linear system associated with the Wronskian:
,
or,
.
  • In first-order logic, the statement , entails the statement . Consequently, statement (2) entails statement (1), and the theorem is proved.