Give an example of a linearly independent set of three funct

Give an example of a linearly independent set of three functions in, F(R, R).

Solution

Let the set \'S\' consist of three functions are defined as follow:

S ={ f₁(x) = x, f₂(x) = x², and f₃(x) = x³ on (,)} and its wronskian W[f₁, f₂, f₃](x) = 2x³,

Since W[f₁, f₂, f₃](x) = 0 (except at x = 0), the functions are linearly independent on (,).