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:
| 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.