A

tool for determining whether a set of

functions is

linearly independent or not.

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:

| x1_{1} x2_{1} x3_{1} ... xn_{1}|
| x1_{2} x2_{2} x3_{2} ... xn_{2}|
W= | x1_{3} x2_{3} x3_{3} ... xn_{3}|
|.....................|
| x1_{n} x2_{n} x3_{n} ... xn_{n}|

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.