0. Thus the set of functions is linearly independent if and only if
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Linearly dependent and independent sets of functions, Wronskian test for dependence
Linear combination of functions. The function c1 f1(x) + c2 f2(x) + ... + cn fn(x) with arbitrary numerical values for the coefficients c1, c2, ... ,cn is called a linear combination of the functions f1(x), f2(x), ... , fn(x).
Linearly dependent and independent sets of functions. A set of functions f1(x), f2(x), ... ,fn(x) is said to be linearly dependent if some one of the functions in the set can be expressed as a linear combination of one or more of the other functions in the set. If none of the functions in the set can be expressed as a linear combination of any other functions of the set, then the set is said to be linearly independent.
Example. The set of four functions x2, 3x + 1, 3x2+ 6x + 2 and x3 is linearly dependent since
3x2+ 6x + 2 = 3(x2) + 2(3x + 1)
A necessary and sufficient condition for the linear independence of a set of functions. There exists an important algebraic criterion, an algebraic test, which can tell us whether a set of functions is linearly independent or not. That test is given by the following theorem:
Theorem. A necessary and sufficient condition for the set of functions f1(x), f2(x), ... ,fn(x) to be linearly independent is that
c1 f1(x) + c2 f2(x) + ... + cn fn(x) = 0
only when all the scalars ci are zero.
What is the reasoning that leads to the assertion of this theorem? Well, a set of functions f1(x), f2(x), ... ,fn(x) is linearly dependent if some one of the functions in the set can be expressed as a linear combination of one or more of the other functions in the set, that is if there exists some function fi(x) in the set such that
fi(x) = a1 fj(x) + a2 fk(x) + ...
for one or more functions fj(x), fk(x) , etc. of the set. This condition implies that there exists some subset of functions fi(x), fj(x), fk(x), etc. within the full set such that
ci fi(x) + cj fj(x) + ck fk(x) + ... = 0
where ci cj, ck, etc. are non-zero. Said differently, a set is linearly dependent if there exist two or more non-zero c’s for which the following equation holds true:
c1 f1(x) + c2 f2(x) + ... + cn fn(x) = 0
(i.e. it is possible for the equation to hold true even though not all of the c’s are zero). If there
does not exist two or more non-zero c’s for which it will hold, then the set of functions is linearly
independent. The case in which only one of the c’s is non-zero is impossible since cixi = 0 is not
possible if c
0. Thus the set of functions is linearly independent if and only if
c1 f1(x) + c2 f2(x) + ... + cn fn(x) = 0
only when all the scalars ci are zero.
Wronskian test for dependence. A test for the linear dependence of a set of n functions f1(x), f2(x), ... , fn(x) having derivatives through the (n-1)th order can be obtained through evaluation of the Wronskian determinant
If the Wronskian is not identically zero, the functions are linearly independent. If it is identically zero over an interval (a, b), the functions are linearly dependent on the interval.
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