Linearly Dependent Functions

The n functions f_1(x), f_2(x), ..., f_n(x) are linearly dependent if, for some c_1, c_2, ..., c_n in R not all zero,


for all x in some interval I. If the functions are not linearly dependent, they are said to be linearly independent. Now, if the functions and in C^(n-1) (the space of functions with n-1 continuous derivatives), we can differentiate (1) up to n-1 times. Therefore, linear dependence also requires


where the sums are over i=1, ..., n. These equations have a nontrivial solution iff the determinant

 |f_1 f_2 ... f_n; f_1^' f_2^' ... f_n^'; | | ... |; f_1^((n-1)) f_2^((n-1)) ... f_n^((n-1))|=0,

where the determinant is conventionally called the Wronskian and is denoted W(f_1,f_2,...,f_n).

If the Wronskian !=0 for any value c in the interval I, then the only solution possible for (2) is c_i=0 (i=1, ..., n), and the functions are linearly independent. If, on the other hand, W=0 over some range, then the functions are linearly dependent somewhere in the range. This is equivalent to stating that if the vectors V[f_1(c)], ..., V[f_n(c)] defined by

 V[f_i(x)]=[f_i(x); f_i^'(x); f_i^('')(x); |; f_i^((n-1))(x)]

are linearly independent for at least one c in I, then the functions f_i are linearly independent in I.

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Sansone, G. "Linearly Independent Functions." §1.2 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 2-3, 1991.

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Linearly Dependent Functions

Cite this as:

Weisstein, Eric W. "Linearly Dependent Functions." From MathWorld--A Wolfram Web Resource.

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