Hello, I am having some trouble proving
that the set {1,x,x^2,x^3,...x^n} is linearly independent in Pn(R).
If where is the zero function. It means that we have a polynomial always equaling 0 for any . It means that that it must be the zero polynomial because otherwise we have a non-zero polynomial having infinitely many zero (for any x) which is impossible (it can have at most n zeros). Thus, have to be all zero.