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Using the inner product of the previous problem, let $\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}$ be an orthogonal basis for the space $\mathcal{P}_{2}$ of quadratic polynomials and let $\mathcal{S}=\operatorname{span}\left(x, x^{2}\right) \subset$ $\mathcal{P}_{2}$. Using the basis $\mathcal{B}$, find the linear map $P: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}$ that is the orthogonal projection from $\mathcal{P}_{2}$ onto $\mathcal{S}$.
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