Legendre's proof involving linearity independence

Question

Show that any polynomial of degree n is a linear combination of P0(x), P1(x), ..., Pn(x)

Actually I have no idea how to start with a proof involving "any". Can someone help??

Answer 1

gram schmidt orthogonalization process under inner product$$ \int _{-1}^{1} f(x)g(x)dx$$ may help.