Chapter 5.3, Problem 61E

Orthonormal Sets in $P_2$ In Exercises 57-62, let $p\left(x\right)=a_0+a_{1}x+a_2{x}^2$ and $q\left(x\right)=b_0+b_{1}x+b_2{x}^2$ be vectors in $P_2$ with $\langle p,q\rangle =a_0{b}_0+a_1{b}_1+a_2{b}_2$. Determine whether the polynomials form an orthonormal set, and if not, apply the Gram-Schmidt orthonormalization process to form an orthonormal set.

$\left\{\frac{1+x^2}{\sqrt{2}},\frac{-1+x+x^2}{\sqrt{3}}\right\}$