| 释义 |
Linearly Dependent FunctionsThe  functions  ,  , ...,  are linearly dependent if, for some  ,  , ...,  not all zero,
 | (1) |
 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) upto  times. Therefore, linear dependence also requires
 | (2) |
 | (3) |
 | (4) |
 , ..., . These equations have a nontrivial solution Iff the Determinant
 | (5) |
 . Ifthe 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, thefunctions are linearly dependent in the range. This is equivalent to stating that if the vectors  ,...,  defined by
 | (6) |
 , 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. |