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Rivi 1:
[[matematiikka|Matematiikassa]] '''Wronskin determinantilla''' tarkoitetaan determinanttia, jonka kehitti {{harvs|txt|authorlink=[[Józef Maria Hoene-Wroński|first=Józef|last=Hoene-Wronski|year=1812}}]] ja nimesi {{harvs|txt|authorlink=[[Thomas Muir (mathematician)|first=Thomas|last=Muir|year=1882|loc=Chapter XVIII}}]]. Sitä käytetään esimerkiksi [[differentiaaliyhtälö]]laskennassa, jossa sen avulla voidaan tarkastella yhtälön ratkaisujen [[lineaarinen riippumattomuus|lineaarista riippumattomuutta]].
 
==Määritelmä==
Rivi 5:
Kahden funktion ''f'' and ''g'' Wronskin determinantti on ''W''(''f'',''g'') = ''fg''′–''gf'' ′.
 
More generally, forYleisemmin ''n'':lle [[real number|realreaaliluku]]- ortai [[complex number|complexkompleksiluku]]-valuedkertoimisille functionsfunktioille ''f<sub>1</sub>'', ..., ''f<sub>n</sub>'', whichjotka areovat ''n''&nbsp;&minus;&nbsp;1 timeskertaa [[differentiablederivoituva|derivoituvia]] on an [[interval (mathematics)väli|intervalvälillä]] ''I'', themuodostetaan Wronskin determinantti Wronskian ''W''(''f''<sub>1</sub>, ..., ''f<sub>n</sub>'') as a function on ''I'' is defined byseuraavasti
 
:<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==
==The Wronskian and linear independence==
 
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''&nbsp;=&nbsp;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.
 
==Generalized Wronskians==
 
For ''n'' functions of several variables, a '''generalized Wronskian''' is the determinant of an ''n'' by ''n'' matrix with entries ''D''<sub>''i''</sub>(''f''<sub>''j''</sub>) (with 0≤i<''n''), where each ''D''<sub>''i''</sub> is some constant coefficient linear partial differential operator of order ''i''. If the functions are linearly dependent then all generalized Wronskians vanish. As in the 1 variable case the converse is not true in general: if all generalized Wronskians vanish this does not imply that the functions are linearly dependent. However the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of [[Thue–Siegel–Roth theorem|Roth's theorem]]. For more general conditions under which the converse is valid see {{harvtxt|Wolsson|1989b}}.
 
==See also==
 
*[[Moore matrix]], analogous to the Wronskian with differentiation replaced by the [[Frobenius endomorphism]] over a finite field.
 
==References==