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??

See one of my other posts on this subject: https://math.stackexchange.com/questions/512295/using-legendre-polynomial-to-approximate-any-polynomial/770500#770500 — Maestro13, Feb 14 '19 at 23:06

score 0 · Answer 1 · answered Jan 18 '19 at 13:52

0

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

answered Jan 18 '19 at 13:52

Bijayan Ray

another approach may be from legendre equation – Bijayan Ray Jan 18 '19 at 13:53
can you elaborate more? – user635977 Jan 19 '19 at 17:20
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent – Bijayan Ray Jan 20 '19 at 12:19

1 Answers1