how to find inner product and linear transformation

Question

1.) $p=p(x)$ and $q=q(x)$ are polinom in $P_2$. Defined by $<p,q>=p(0)q(a)+p(\frac{1}{2})$$q(b)+p(1)q(c)$ for certain $a,b,c$ such that the $<.,.>$ is inner product in$P_2$ . Find the value of $\left\lVert 4x^2-1\right\rVert$

I can not determine the value of a,b,c such that <,> is inner product

2.) $T_1 : R^(3x3)$ -> R and $T_2 : R^(3x3)->R^(3x3)$ are linear transformation defined by $T_2(A)=A^T$. If $A=$\begin{matrix} a & b & c \\d & e & f \\g & h & i \\ \end{matrix} and $(T_1(T_2(A))=T_1(A)$ , find $T_1(A)$

$T_1(A)=T_1(A^T)$ using this fact how can we do for solving this question

Answer 1

$$\left\|4x^2-1\right\|^2=\langle 4x^2-1,\,4x^2-1\rangle=4\langle x^2,\,x^2\rangle-8\langle x^2,\,1\rangle+\langle 1,\,1\rangle$$

and now apply the definition. For example, taking $\;q(x):=x^2\;$ , we get

$$\langle x^2,\,x^2\rangle=q(0)q(a)+q\left(\frac12\right)q(b)+q(1)q(c)=\frac{b^2}4+c^2$$

and etc.