The
functions ,
,
...,
are linearly dependent if, for some , , ..., not all zero,

(1)

for all
in some interval . If the functions are not linearly dependent, they are said
to be linearly independent. Now, if the functions and in (the space of functions with continuous derivatives), we can differentiate (1)
up to
times. Therefore, linear dependence also requires

(2)

(3)

(4)

where the sums are over , ..., . These equations have a nontrivial solution iff
the determinant

If the Wronskian for any value in the interval , then the only solution possible for (2) is (, ..., ), and the functions are linearly independent. If, on the other
hand,
over some range, then the functions are linearly dependent somewhere in the range.
This is equivalent to stating that if the vectors , ..., defined by

(6)

are linearly independent for at least one , then the functions are linearly independent in .