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 $S := \{ p \in \mathcal{P}3 | p(t) = t + c, -1 \le c \le 1 \}$.

 (a) Describe $S^{\perp}$ as the span of a set of its basis composed of some monic polynomials, i.e. polynomials with 1 as the coefficient of their highest degrees.