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Linearly dependent and independent sets of functions, Wronskian test for dependence

Linear combination of functions. The function c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) _{ } with
arbitrary numerical values for the coefficients c_{1}, c_{2}, ... ,c_{n} is called a linear combination of the
functions f_{1}(x), f_{2}(x), ... , f_{n}(x).

Linearly dependent and independent sets of functions. A set of functions f_{1}(x),
f_{2}(x), ... ,f_{n}(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 x^{2}, 3x + 1, 3x^{2}+ 6x + 2 and x^{3} is linearly dependent since

3x^{2}+ 6x + 2 = 3(x^{2}) + 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 f_{1}(x), f_{2}(x), ...
,f_{n}(x) _{ }to be linearly independent is that

c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) = 0

only when all the scalars c_{i} are zero.

What is the reasoning that leads to the assertion of this theorem? Well, a set of functions f_{1}(x),
f_{2}(x), ... ,f_{n}(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 f_{i}(x) in the set such that

f_{i}(x) = a_{1} f_{j}(x) + a_{2} f_{k}(x) + ...

for one or more functions f_{j}(x), f_{k}(x) , etc. of the set. This condition implies that there exists
some subset of functions f_{i}(x), f_{j}(x), f_{k}(x), etc. within the full set such that

c_{i }f_{i}(x) + c_{j} f_{j}(x) + c_{k} f_{k}(x) + ... = 0

where c_{i} c_{j}, c_{k}, 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:

c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(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 c_{i}x_{i} = 0 is not
possible if c
0. Thus the set of functions is linearly independent if and only if

c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) = 0

only when all the scalars c_{i} are zero.

Wronskian test for dependence. A test for the linear dependence of a set of n functions
f_{1}(x), f_{2}(x), ... , f_{n}(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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