The functions , , ..., are linearly dependent if, for some , , ...,
not all zero,
(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
where the sums are over , ..., . These equations have a nontrivial solution Iff the Determinant
where the Determinant is conventionally called the Wronskian and is denoted
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
are linearly independent for at least one , then the functions are linearly independent in .
Sansone, G. ``Linearly Independent Functions.'' §1.2 in Orthogonal Functions, rev. English ed.
New York: Dover, pp. 2-3, 1991.
© 1996-9 Eric W. Weisstein