Question

Exercise 3-8 refer to \({{\bf{P}}_{\bf{2}}}\) with the inner product given by evaluation at \( - {\bf{1}}\), 0, and 1. (See Example 2).

6. Compute \(\left\| p \right\|\) and \(\left\| q \right\|\), for p and q in Exercise 4.

Short answer

The values are \(\left\| p \right\| = \sqrt {20} \), and \(\left\| q \right\| = \sqrt {59} \).

Step-by-step solution

Write the results from Exercise 4

\(\begin{align*}p\left( { - 1} \right) &= 3\left( { - 1} \right) - {\left( { - 1} \right)^2}\\ &= - 3 - 1\\ &= - 4\end{align*}\)

\(\begin{align*}p\left( 0 \right) &= 3\left( 0 \right) - {\left( 0 \right)^2}\\ &= 0\end{align*}\)

\(\begin{align*}p\left( 1 \right) &= 3\left( 1 \right) - {\left( 1 \right)^2}\\ &= 3 - 1\\ &= 2\end{align*}\)

And,

\(\begin{align*}q\left( { - 1} \right) &= 3 + 2{\left( { - 1} \right)^2}\\ &= 5\end{align*}\)

\(\begin{align*}q\left( 0 \right) &= 3 + 2{\left( 0 \right)^2}\\ &= 3\end{align*}\)

\(\begin{align*}q\left( 1 \right) &= 3 + 2{\left( 1 \right)^2}\\ &= 5\end{align*}\)

Find the value of \(\left\| p \right\|\)

The value of \(\left\| p \right\|\) can be calculated as follows:

\(\begin{align*}\left\| p \right\| &= \sqrt {\left\langle {p,p} \right\rangle } \\ &= \sqrt {p\left( { - 1} \right)p\left( { - 1} \right) + p\left( 0 \right)p\left( 0 \right) + p\left( 1 \right)p\left( 1 \right)} \\ &= \sqrt {\left( { - 4} \right)\left( { - 4} \right) + \left( 0 \right)\left( 0 \right) + \left( 2 \right)\left( 2 \right)} \\ &= \sqrt {20} \end{align*}\)

Thus, the value of \(\left\| p \right\|\) is \(\sqrt {20} \).

Find the value of \(\left\| q \right\|\)

The value of \(\left\| q \right\|\) can be calculated as follows:

\(\begin{align*}\left\| q \right\| &= \sqrt {\left\langle {q,q} \right\rangle } \\ &= \sqrt {q\left( { - 1} \right)q\left( { - 1} \right) + q\left( 0 \right)q\left( 0 \right) + q\left( 1 \right)q\left( 1 \right)} \\ &= \sqrt {\left( 5 \right)\left( 5 \right) + \left( 3 \right)\left( 3 \right) + \left( 5 \right)5} \\ &= \sqrt {59} \end{align*}\)

Thus, the value of \(\left\| q \right\|\) is \(\sqrt {59} \).