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Rivi 1:
[[matematiikka|Matematiikassa]] '''Wronskin determinantilla''' tarkoitetaan determinanttia, jonka kehitti
==Määritelmä==
Rivi 5:
Kahden funktion ''f'' and ''g'' Wronskin determinantti on ''W''(''f'',''g'') = ''fg''′–''gf'' ′.
:<math>
Rivi 21:
When the functions ''f''<sub>''i''</sub> are solutions of a [[linear differential equation]], the Wronskian can be found explicitly using [[Abel's identity]], even if the functions ''f''<sub>''i''</sub> are not known explicitly.
==Wronskin determinantti ja lineaarinen riippumattomuus==
If the functions ''f''<sub>''i''</sub> are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. So the Wronskian can be used to show that a set of differentiable functions is [[linear independence|linearly independent]] on an interval by showing that it does not vanish identically.
Rivi 27:
A common misconception is that ''W'' = 0 everywhere implies linear dependence, but {{harvtxt|Peano|1889}} pointed out that the functions ''x''<sup>2</sup> and |''x''|''x'' have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in a neighborhood of 0. There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence.
{{harvtxt|Peano|1889}} observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent. {{harvtxt|Bochner|1901}} gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of ''n'' functions is identically zero and the ''n'' Wronskians of ''n''–1 of them do not all vanish at any point then the functions are linearly dependent. {{harvtxt|Wolsson|1989a}} gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.
==References==
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