Question

For certain polynomials \(\mathbf{p}(t), \mathbf{q}(t)\), and \(\mathbf{r}(t)\), say we are given the following table of inner products:

\begin{align}
\begin{array}{|c||c|c|c|}
\hline\langle,\rangle & \mathbf{p} & \mathbf{q} & \mathbf{r} \\
\hline \hline \mathbf{p} & 4 & 0 & 8 \\
\mathbf{q} & 0 & 1 & 0 \\
\mathbf{r} & 8 & 0 & 50 \\
\hline
\end{array}
\end{align}

For example, \(\langle\mathbf{q}, \mathbf{r}\rangle=\langle\mathbf{r}, \mathbf{q}\rangle=0\). Let \(E\) be the span of \(\mathbf{p}\) and \(\mathbf{q}\).
a) Compute \(\langle\mathbf{p}, \mathbf{q}+\mathbf{r}\rangle\).
b) Compute \(\|\mathbf{q}+\mathbf{r}\|\).
c) Find the orthogonal projection \(\operatorname{Proj}_{E} \mathbf{r}\). [Express your solution as linear combinations of \(\mathbf{p}\) and \(\mathbf{q}\).]
d) Find an orthonormal basis of the span of \(\mathbf{p}, \mathbf{q}\), and \(\mathbf{r}\). [Express your results as linear combinations of \(\mathbf{p}, \mathbf{q}\), and \(\mathbf{r}\).]