(Steps Shown) The first three Legendre polynomials are P_0(x)=1, P_1(x)=x, P_2(x)=1/2(3x^2-1) Show that the orthonormal vectors in L^2(-1,1) obtained by


Question: The first three Legendre polynomials are

\[{{P}_{0}}\left( x \right)=1,\,\,{{P}_{1}}\left( x \right)=x,\,\,\,{{P}_{2}}\left( x \right)=\frac{1}{2}\left( 3{{x}^{2}}-1 \right)\]

Show that the orthonormal vectors in \({{L}^{2}}\left( -1,1 \right)\) obtained by applying the Gram-Schmidt process to \(1,x,{{x}^{2}}\) are scalar multiples of these.

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