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Chapter 6: Q-6-6SE (page 331)
Show that if \(U\) is an orthogonal matrix, then any real eigenvalue of \(U\) must be \( \pm 1\).
It is proved that any real eigenvalues of \(U\) is \( \pm 1\).
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In exercises 7-10, show that {u1, u2} or {u1,u2,u3} is an orthogonal basis for \({\mathbb{R}^2}\) or \({\mathbb{R}^3}\), respectively. Then express x as a linear combination of the u.
7. \[{u_1} = \left[ {\begin{align}2\\{ - 3}\end{align}} \right]\], \[{u_2} = \left[ {\begin{align}6\\4\end{align}} \right]\], and \[x = \left[ {\begin{align}9\\{ - 7}\end{align}} \right]\]
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 \(\left( { - {\bf{1}},{\bf{2}},{\bf{0}}, - {\bf{2}},{\bf{1}}} \right)\).
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.\[y = \left[ {\begin{aligned}3\\1\\5\\1\end{aligned}} \right]\],\[{{\bf{v}}_1} = \left[ {\begin{aligned}3\\1\\{ - 1}\\1\end{aligned}} \right]\],\[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\{ - 1}\\1\\{ - 1}\end{aligned}} \right]\]
According to Kepler鈥檚 first law, a comet should have an elliptic, parabolic, or hyperbolic orbit (with gravitational attractions from the planets ignored). In suitable polar coordinates, the position \(\left( {r,\vartheta } \right)\) of a comet satisfies an equation of the form
\(r = \beta + e\left( {r \cdot \cos \vartheta } \right)\)
Where \(\beta \) is a constant and \(e\) is the eccentricity of the orbit, with \(0 \le e < 1\) for an ellipse, \(e = 1\) for a parabola, and \(e > 1\) for a hyperbola. Suppose observations of a newly discovered comet provide the data below. Determine the type of orbit, and predict where the comet will be when \(\vartheta = 4.6\left( {{\rm{radians}}} \right)\).
\(\begin{array}{{}{}} \vartheta & & {.88}&{1.10}&{1.42}&{1.77}&{2.14} \\ \hline r& {3.00}&{2.30}&{1.65}&{1.25}&{1.01} \end{array}\)
In Exercises 3鈥�6, verify that\[\left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\]is an orthogonal set, and then find the orthogonal projection of\[{\bf{y}}\]onto Span\[\left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\].
5.\[y = \left[ {\begin{aligned}{ - 1}\\2\\6\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}3\\{ - 1}\\2\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}1\\{ - 1}\\{ - 2}\end{aligned}} \right]\]
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