The problem reads as follows:
Look at the inner product in $\mathbf{P}_2$ given by $\langle p,q \rangle = p(0)q(0) + p(\frac{1}{2})q(\frac{1}{2}) + p(1)q(1)$.
Find an orthogonal basis $\mathbf{C}$ for $\mathbf{P}_2$.
Now, I know I can use the standard basis for $\mathbf{P}_2$, i.e. $B=\{1,t,t^2 \}$ and apply Gram-Schmidt. So: $\mathbf{v}_1 = \mathbf{x}_1 = 1$
$\mathbf{v}_2 = \mathbf{x}_2 - proj_{\mathbf{v}_1} \mathbf{x}_2 = \mathbf{x}_2 - \frac{\langle \mathbf{x}_2, \mathbf{v}_1 \rangle}{\langle \mathbf{v}_1, \mathbf{v}_1 \rangle}\mathbf{v}_1 $ and so on.
My question is incredibly trivial: I just can't figure out how to calculate the inner products in the equation above using the one in the problem.
From a solution of the problem I know that $\langle \mathbf{v}_1, \mathbf{v}_1 \rangle = 1$ and $\langle \mathbf{x}_2, \mathbf{v}_1 \rangle = \langle t,1\rangle= 3/2$. Could someone please show me the calculations, or how to insert these into the equation in the problem?