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Chapter 6: Q12E (page 331)

Find a polynomial \({p_{\bf{3}}}\) such that \(\left\{ {{p_{\bf{0}}},{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\) (see Exercise 11) is an orthogonal basis for the subspace \({{\bf{P}}_{\bf{3}}}\) of \({{\bf{P}}_{\bf{4}}}\). Scale the polynomials \({p_{\bf{3}}}\) so that vector of values is \(\left( { - {\bf{1}},{\bf{2}},{\bf{0}}, - {\bf{2}},{\bf{1}}} \right)\).

Short Answer

Expert verified

The vector \({p_3}\) is \(\frac{5}{6}\left( {{t^3} - \frac{{17}}{5}t} \right)\).

Step by step solution

01

Find the vector \({p_{\bf{3}}}\)

Let the subspace W is defined as:

\(W = {\rm{Span}}\left\{ {{p_0},{p_1},{p_2}} \right\}\)

The vector \({p_3}\) is:

\(\begin{aligned}{p_3} &= p - {\rm{pro}}{{\rm{j}}_W}p\\ &= {t^3} - \frac{{17}}{5}t\end{aligned}\)

Thus, \({p_3}\) makes \(\left\{ {{p_0},{p_1},{p_2},{p_3}} \right\}\) and orthogonal basis for the subspace \({{\bf{P}}_3}\) and \({{\bf{P}}_4}\).

02

Find the values of \({p_{\bf{3}}}\) 

The values of \({p_3}\) are:

\(\begin{aligned}{p_3}\left( { - 2} \right) &= - \frac{6}{5}\\{p_3}\left( { - 1} \right) &= \frac{{12}}{5}\\{p_3}\left( 0 \right) &= 0\\{p_3}\left( 1 \right) &= - \frac{{12}}{5}\\{p_3}\left( 2 \right) &= \frac{6}{5}\end{aligned}\)

So, scaling the vector by \(\frac{5}{6}\), the vector \({p_3}\) can be expressed as \({p_3} = \frac{5}{6}\left( {{t^3} - \frac{{17}}{5}t} \right)\).

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Most popular questions from this chapter

Given data for a least-squares problem, \(\left( {{x_1},{y_1}} \right), \ldots ,\left( {{x_n},{y_n}} \right)\), the following abbreviations are helpful:

\(\begin{aligned}{l}\sum x = \sum\nolimits_{i = 1}^n {{x_i}} ,{\rm{ }}\sum {{x^2}} = \sum\nolimits_{i = 1}^n {x_i^2} ,\\\sum y = \sum\nolimits_{i = 1}^n {{y_i}} ,{\rm{ }}\sum {xy} = \sum\nolimits_{i = 1}^n {{x_i}{y_i}} \end{aligned}\)

The normal equations for a least-squares line \(y = {\hat \beta _0} + {\hat \beta _1}x\) may be written in the form

\(\begin{aligned}{c}{{\hat \beta }_0} + {{\hat \beta }_1}\sum x = \sum y \\{{\hat \beta }_0}\sum x + {{\hat \beta }_1}\sum {{x^2}} = \sum {xy} {\rm{ (7)}}\end{aligned}\)

Derive the normal equations (7) from the matrix form given in this section.

[M] Let \({f_{\bf{4}}}\) and \({f_{\bf{5}}}\) be the fourth-order and fifth order Fourier approximations in \(C\left[ {{\bf{0}},{\bf{2}}\pi } \right]\) to the square wave function in Exercise 10. Produce separate graphs of \({f_{\bf{4}}}\) and \({f_{\bf{5}}}\) on the interval \(\left[ {{\bf{0}},{\bf{2}}\pi } \right]\), and produce graph of \({f_{\bf{5}}}\) on \(\left[ { - {\bf{2}}\pi ,{\bf{2}}\pi } \right]\).

In Exercises 7鈥�10, let\[W\]be the subspace spanned by the\[{\bf{u}}\]鈥檚, and write y as the sum of a vector in\[W\]and a vector orthogonal to\[W\].

8.\[y = \left[ {\begin{aligned}{ - 1}\\4\\3\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}1\\1\\{\bf{1}}\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 1}\\3\\{ - 2}\end{aligned}} \right]\]

Question: In Exercises 3-6, verify that\(\left\{ {{{\bf{u}}_{\bf{1}}},{{\bf{u}}_{\bf{2}}}} \right\}\)is an orthogonal set, and then find the orthogonal projection of y onto\({\bf{Span}}\left\{ {{{\bf{u}}_{\bf{1}}},{{\bf{u}}_{\bf{2}}}} \right\}\).

3.\[y = \left[ {\begin{aligned}{ - {\bf{1}}}\\{\bf{4}}\\{\bf{3}}\end{aligned}} \right]\],\({{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{\bf{1}}\\{\bf{1}}\\{\bf{0}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{ - {\bf{1}}}\\{\bf{1}}\\{\bf{0}}\end{aligned}} \right]\)

a. Rewrite the data in Example 1 with new \(x\)-coordinates in mean deviation form. Let \(X\) be the associated design matrix. Why are the columns of \(X\) orthogonal?

b. Write the normal equations for the data in part (a), and solve them to find the least-squares line, \(y = {\beta _0} + {\beta _1}x*\), where \(x* = x - 5.5\).
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