Let $p_0$ and $p_1$ be the polynomials in $\mathbb P_2$ given below, and let the inner product for $\mathbb P_2$ be given by evaluation at points $-2, 0, 2$, so $$ \langle p,q\rangle = p(−2)q(−2)+p(0)q(0)+p(2)q(2). $$ Find the inner product of $p_0$ and $p_1$ and the norms of $p_0$ and $p_1$. Use the square root symbol $\sqrt{}$ where needed to give an exact value for your answer.
$$p_0 = −2x+1,\\ p_1 = −2x^2+2x−2.$$
As I understood the inner product is basically the dot product, I would think the answer to the following would be
$$ \langle p_0, p1\rangle = −6, \\ \|p_0\| = \sqrt 5 ,\\ \|p_1 \| = \sqrt 12 .$$
but I have seen that the answers are wrong.
First the inner product. Using the definition we find
$$\langle p_0,p_1\rangle = p_0(-2)p_1(-2) + p_0(0)p_1(0) + p_0(2)p_1(2) = \\ =5(-14) + 1(-2) + (-3)(-6)= -70 -2 +18 = -54 . $$
The norms are compited in the same way:
$$||p_0|| = \sqrt{\langle p_0,p_0\rangle} = \sqrt{5^2 + 1^2 + (-3)^2} = \sqrt{35} ,\\ ||p_1|| = \sqrt{\langle p_1,p_1\rangle} =\sqrt{14^2 + (-2)^2 + (-6)^2} = \sqrt{236} . $$