## Linearly Dependent Functions

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. 2-3, 1991.