Wronskian (idea)

A tool for determining whether a set of functions is linearly independent or not.

For Functions of One Variable:

For a given set of functions {g1_(x), g2_(x), g3_(x) ... gn_(x)} the Wronskian is defined by

  
	| g1     g2      g3   ...   gn    |
	| g'1    g'2     g'3  ...   g'n   |
W=	| g''1   g''2    g''3 ...   g''n  |
	| ................................|
	| g(n-1)1 g(n-1)2 g(n-1)3 ... g(n-1)n|

Which is interpreted as the determinant of the square matrix formed by n rows, the first row consisting of the functions in question, the second row consisting of their first derivatives, the third row consisting of their second derivatives, and so on, up to the nth row consisting of their (n-1) derivatives.

If the Wronskian is not equal to zero for any value x in the domain of {g1, g2, g3...gn} then the functions are linearly independent. The converse is also true. If W = 0 for all x in the domain, then the functions are linearly dependent.

It is also possible to determine if a set of functions is linearly independent on a given interval by considering only values of x in that interval. If W = 0 for all x in an interval I, then the set of functions is linearly dependent on I.

_{For information on how to evaluate this determinant, see determinant}

For Vector Functions:

For a set of n column vectors {x1_(t), x2_(t), x3_(t) ... xn_(t)}, each with n elements, the Wronskian is defined by:

  
	| x11  x21  x31 ... xn1|
	| x12  x22  x32 ... xn2|
W=	| x13  x23  x33 ... xn3|
	|.....................|
	| x1n  x2n  x3n ... xnn|

In this case, the Wronskian is simply the determinant of the matrix formed by combining the individual column vectors. (Note however, that there must be n column vectors each with n rows because the determinant is only defined for square matrices.)

The same rules for determining linear independence or dependence apply as for functions of one variable. If W is nonzero at any point t on an interval I, the set of vectors is linearly independent on I.