Linearly Dependent Functions

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

(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, 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 .