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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??

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

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  • $\begingroup$ another approach may be from legendre equation $\endgroup$ – Bijayan Ray Jan 18 at 13:53
  • $\begingroup$ can you elaborate more? $\endgroup$ – user635977 Jan 19 at 17:20
  • $\begingroup$ 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 $\endgroup$ – Bijayan Ray Jan 20 at 12:19

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