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Chapter 6: Q9E (page 331)

Let \({{\bf{P}}_{\bf{3}}}\) have the inner product given by evaluation at \( - {\bf{3}}\), \( - {\bf{1}}\),1, and 3. Let \({p_{\bf{0}}}\left( t \right) = {\bf{1}}\), \({p_{\bf{1}}}\left( t \right) = t\), and \({p_{\bf{2}}}\left( t \right) = {t^{\bf{2}}}\).

  1. Compute the orthogonal projection of \({p_{\bf{2}}}\) onto the sub-spaced spanned by , and \({p_{\bf{1}}}\).
  2. Find the polynomial q that is orthogonal to \({p_{\bf{0}}}\) and , such that \(\left\{ {{p_{\bf{0}}},{p_{\bf{1}}},q} \right\}\) is an orthogonal basis for . Scale the polynomial q so that its vector of values at \(\left( { - {\bf{3}}, - {\bf{1}},{\bf{1}},{\bf{3}}} \right)\) is \(\left( {{\bf{1}}, - {\bf{1}}, - {\bf{1}},{\bf{1}}} \right)\).

Short Answer

Expert verified

a. 5

b. \(\frac{1}{4}\left( {{t^2} - 5} \right)\)

Step by step solution

01

Find the values of polynomials

The value of \({p_0}\left( t \right)\) is 1 for all values of \(t\).

The values of \({p_1}\left( t \right) = t\) are:

\(\begin{aligned}{p_1}\left( { - 3} \right) = - 3\\{p_1}\left( { - 1} \right) = - 1\\{p_1}\left( 1 \right) = 1\\{p_1}\left( 3 \right) = 3\end{aligned}\)

The values of \({p_2}\left( t \right) = {t^2}\) are:

\(\begin{aligned}{p_2}\left( { - 3} \right) = 9\\{p_2}\left( { - 1} \right) = 1\\{p_2}\left( 1 \right) = 1\\{p_2}\left( 3 \right) = 9\end{aligned}\)

02

Find the inner products

Find the inner product \(\left\langle {{p_2},{p_0}} \right\rangle \).

\(\begin{aligned}\left\langle {{p_2},{p_0}} \right\rangle &= {p_2}\left( { - 3} \right){p_0}\left( { - 3} \right) + {p_2}\left( { - 1} \right){p_0}\left( { - 1} \right) + {p_2}\left( 1 \right){p_0}\left( 1 \right) + {p_2}\left( 3 \right){p_0}\left( 3 \right)\\ &= \left( 9 \right)\left( 1 \right) + \left( 1 \right)\left( 1 \right) + \left( 1 \right)\left( 1 \right) + \left( 9 \right)\left( 1 \right)\\ &= 20\end{aligned}\)

Find the inner product \(\left\langle {{p_2},{p_1}} \right\rangle \).

\(\begin{aligned}\left\langle {{p_2},{p_1}} \right\rangle &= {p_2}\left( { - 3} \right){p_1}\left( { - 3} \right) + {p_2}\left( { - 1} \right){p_1}\left( { - 1} \right) + {p_2}\left( 1 \right){p_1}\left( 1 \right) + {p_2}\left( 3 \right){p_1}\left( 3 \right)\\ &= \left( 9 \right)\left( { - 3} \right) + \left( 1 \right)\left( { - 1} \right) + \left( 1 \right)\left( 1 \right) + \left( 9 \right)\left( 3 \right)\\ &= 0\end{aligned}\)

Find the inner product \(\left\langle {{p_0},{p_0}} \right\rangle \).

\(\begin{aligned}\left\langle {{p_0},{p_0}} \right\rangle &= {p_0}\left( { - 3} \right){p_0}\left( { - 3} \right) + {p_0}\left( { - 1} \right){p_0}\left( { - 1} \right) + {p_0}\left( 1 \right){p_0}\left( 1 \right) + {p_0}\left( 3 \right){p_1}\left( 3 \right)\\ &= 1 + 1 + 1 + 1\\ &= 4\end{aligned}\)

Find the inner product \(\left\langle {{p_1},{p_1}} \right\rangle \).

\(\begin{aligned}\left\langle {{p_1},{p_1}} \right\rangle &= {p_1}\left( { - 3} \right){p_1}\left( { - 3} \right) + {p_1}\left( { - 1} \right){p_1}\left( { - 1} \right) + {p_1}\left( 1 \right){p_1}\left( 1 \right) + {p_1}\left( 3 \right){p_1}\left( 3 \right)\\ &= \left( { - 3} \right)\left( { - 3} \right) + \left( { - 1} \right)\left( { - 1} \right) + \left( 1 \right)\left( 1 \right) + \left( 3 \right)\left( 3 \right)\\ &= 20\end{aligned}\)

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Most popular questions from this chapter

Compute the quantities in Exercises 1-8 using the vectors

\({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 1}\\2\end{aligned}} \right),{\rm{ }}{\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}4\\6\end{aligned}} \right),{\rm{ }}{\mathop{\rm w}\nolimits} = \left( {\begin{aligned}{*{20}{c}}3\\{ - 1}\\{ - 5}\end{aligned}} \right),{\rm{ }}{\mathop{\rm x}\nolimits} = \left( {\begin{aligned}{*{20}{c}}6\\{ - 2}\\3\end{aligned}} \right)\)

  1. \({\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} ,{\rm{ }}{\mathop{\rm v}\nolimits} \cdot {\mathop{\rm u}\nolimits} ,\,\,{\mathop{\rm and}\nolimits} \,\,\frac{{{\mathop{\rm v}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}{{{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm u}\nolimits} }}\)

A certain experiment produce the data \(\left( {1,7.9} \right),\left( {2,5.4} \right)\) and \(\left( {3, - .9} \right)\). Describe the model that produces a least-squares fit of these points by a function of the form

\(y = A\cos x + B\sin x\)

Find an orthogonal basis for the column space of each matrix in Exercises 9-12.

9. \(\left[ {\begin{aligned}{{}{}}3&{ - 5}&1\\1&1&1\\{ - 1}&5&{ - 2}\\3&{ - 7}&8\end{aligned}} \right]\)

In Exercises 11 and 12, find the closest point to\[{\bf{y}}\]in the subspace\[W\]spanned by\[{{\bf{v}}_1}\], and\[{{\bf{v}}_2}\].

11.\[y = \left[ {\begin{aligned}3\\1\\5\\1\end{aligned}} \right]\],\[{{\bf{v}}_1} = \left[ {\begin{aligned}3\\1\\{ - 1}\\1\end{aligned}} \right]\],\[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\{ - 1}\\1\\{ - 1}\end{aligned}} \right]\]

Find the distance between \({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}0\\{ - 5}\\2\end{aligned}} \right)\) and \({\mathop{\rm z}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 4}\\{ - 1}\\8\end{aligned}} \right)\).
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