6. Consider $V = \def\P{\mathbb P^2}\P$ with inner product: $$\def\sp#1{\left<#1\right>}\sp{p(X), q(X)} = 2p(-1)q(-1) + 3p(1)q(1) + p(2)q(2) $$
a. Show that for any non-zero polynomial $p(X) \in \P$, $\sp{p(X), p(X)} > 0$.
b. Let $W = {\rm span}\{X, X^2\}$. Find an orthonormal basis for $W$ (under the inner product above) using the Gram-Schmidt Algorithm.
c. Express the polynomial $p(X) = X^2 - 2X + 3$ as a sum $p(X) = w_1(X) + w_2(X)$ where $w_1(X) \in W$ and $w_2(X) \in W^\bot$.
d. From your answers in (c) find a basis for $W^\bot$.
I have no idea how to solve this problem. please help! Thanks in advance!
For (a), this boils down to $2(p(-1)^2)+3(p(1))^2+(p(2))^2$. Why must this be $>0$ if $p$ is a nonzero polynomial? Hint: this would not be true in $\mathbb P^3$.
(b) should be pretty straightforward if you follow the algorithm.
For (c), since $W=\text{Span}(x,x^2)$, the polynomial $w_1(x)=x^2-2x\in W$. Can you prove that $p(x)-w_1(x)$ is in $W^\perp$?
For (d), use $w_2(x)$ from (c) to find a basis. What is the dimension of $W^\perp$?