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

(1) 
(where Einstein Summation is used) for all in some interval . If the functions are not linearly dependent,
they are said to be linearly independent. Now, if the functions
, 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

(5) 
where the Determinant is conventionally called the Wronskian and is denoted
. 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, for a range, the
functions are linearly dependent 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 .
References
Sansone, G. ``Linearly Independent Functions.'' §1.2 in Orthogonal Functions, rev. English ed.
New York: Dover, pp. 23, 1991.
© 19969 Eric W. Weisstein
19990525