申論題

15. Consider the vector space $\mathcal{P}3$, the set of all real coefficient polynomials of degree less than 3, the inner product $\langle f, g \rangle := \int{-1}^1 f(t)g(t) dt$ for any $f, g \in \mathcal{P}3$. Denote $S := \{ p \in \mathcal{P}3 | p(t) = t + c, -1 \le c \le 1 \}$.

(c) What is the orthogonal projection of $p(t) := t^2 - t + 1$ onto the subspace $(S^{\perp})^{\perp}$?