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Chapter 6: Q31SE (page 331)
Show that if \(x\) is in both \(W\) and \({W^ \bot }\), then \(x = 0\).
It is proved that \(x = 0\).
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Question: In exercises 1-6, determine which sets of vectors are orthogonal.
\(\left[ {\begin{align}1\\{ - 2}\\1\end{align}} \right]\), \(\left[ {\begin{align}0\\1\\2\end{align}} \right]\), \(\left[ {\begin{align}{ - 5}\\{ - 2}\\1\end{align}} \right]\)
To measure the take-off performance of an airplane, the horizontal position of the plane was measured every second, from \(t = 0\) to \(t = 12\). The positions (in feet) were: 0, 8.8, 29.9, 62.0, 104.7, 159.1, 222.0, 294.5, 380.4, 471.1, 571.7, 686.8, 809.2.
a. Find the least-squares cubic curve \(y = {\beta _0} + {\beta _1}t + {\beta _2}{t^2} + {\beta _3}{t^3}\) for these data.
b. Use the result of part (a) to estimate the velocity of the plane when \(t = 4.5\) seconds.
In Exercises 5 and 6, describe all least squares solutions of the equation \(A{\bf{x}} = {\bf{b}}\).
5. \(A = \left( {\begin{aligned}{{}{}}{\bf{1}}&{\bf{1}}&{\bf{0}}\\{\bf{1}}&{\bf{1}}&{\bf{0}}\\{\bf{1}}&{\bf{0}}&{\bf{1}}\\{\bf{1}}&{\bf{0}}&{\bf{1}}\end{aligned}} \right)\), \({\bf{b}} = \left( {\begin{aligned}{{}{}}{\bf{1}}\\{\bf{3}}\\{\bf{8}}\\{\bf{2}}\end{aligned}} \right)\)
Let \({\mathbb{R}^{\bf{2}}}\) have the inner product of Example 1, and let \({\bf{x}} = \left( {{\bf{1}},{\bf{1}}} \right)\) and \({\bf{y}} = \left( {{\bf{5}}, - {\bf{1}}} \right)\).
a. Find\(\left\| {\bf{x}} \right\|\),\(\left\| {\bf{y}} \right\|\), and\({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).
b. Describe all vectors\(\left( {{z_{\bf{1}}},{z_{\bf{2}}}} \right)\), that are orthogonal to y.
In Exercises 9-12, find a unit vector in the direction of the given vector.
12. \(\left( {\begin{array}{*{20}{c}}{\frac{8}{3}}\\2\end{array}} \right)\)
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