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Chapter 6: Q2E (page 331)

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{1}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{2}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}\\{\bf{7}}\end{aligned}} \right)\)

Short Answer

Expert verified

The affine combination is \({\bf{y}} = - 5{{\bf{v}}_1} + 2{{\bf{v}}_2} + 4{{\bf{v}}_3}\).

Step by step solution

01

Find the translated point

Write the translated points as shown below:

\({{\bf{v}}_2} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}{ - 2}\\1\end{aligned}} \right)\)

\({{\bf{v}}_3} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}2\\1\end{aligned}} \right)\)

\({\bf{y}} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}4\\6\end{aligned}} \right)\)

Write the equation by using the translated matrix as shown below:

\(\begin{aligned}{c}{\bf{y}} - {{\bf{v}}_1} = {c_2}\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + {c_3}\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\\\left( {\begin{aligned}{*{20}{c}}4\\6\end{aligned}} \right) = {c_2}\left( {\begin{aligned}{*{20}{c}}{ - 2}\\1\end{aligned}} \right) + {c_3}\left( {\begin{aligned}{*{20}{c}}2\\1\end{aligned}} \right)\end{aligned}\)

02

Write the augmented matrix

The augmented matrix can be written as shown below:

\(M = \left( {\begin{aligned}{*{20}{c}}{ - 2}&2&4\\1&1&6\end{aligned}} \right)\)

Row reduce the augmented matrix as shown below:

\(\begin{aligned}{c}M = \left( {\begin{aligned}{*{20}{c}}{ - 2}&2&4\\1&1&6\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&{ - 1}&{ - 2}\\1&1&6\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{R_1} \to - \frac{1}{2}{R_1}} \right)\,\\ = \left( {\begin{aligned}{*{20}{c}}1&{ - 1}&{ - 2}\\0&2&8\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{R_1} \to {R_2} - {R_1}} \right)\,\\ = \left( {\begin{aligned}{*{20}{c}}1&{ - 1}&{ - 2}\\0&1&4\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{R_2} \to \frac{1}{2}{R_2}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&0&2\\0&1&4\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{R_1} \to {R_1} + {R_2}} \right)\end{aligned}\)

03

Write the system of equations

From the augmented matrix, the system of equation is shown below:

\({c_2} = 2\)

And,

\({c_3} = 4\)

Substitute the value in the equation of translated points as shown below:

\(\begin{aligned}{c}{\bf{y}} - {{\bf{v}}_1} &= 2\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + 4\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\\{\bf{y}} &= {{\bf{v}}_1} + 2{{\bf{v}}_2} - 2{{\bf{v}}_1} + 4{{\bf{v}}_3} - 4{{\bf{v}}_1}\\{\bf{y}} &= - 5{{\bf{v}}_1} + 2{{\bf{v}}_2} + 4{{\bf{v}}_3}\end{aligned}\)

So, the vector \({\bf{y}}\) is \( - 5{{\bf{v}}_1} + 2{{\bf{v}}_2} + 4{{\bf{v}}_3}\).

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Most popular questions from this chapter

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\[W\].

10.\[y = \left[ {\begin{aligned}3\\4\\5\\6\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}1\\1\\0\\{ - 1}\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}1\\0\\1\\1\end{aligned}} \right]\],\[{{\bf{u}}_3} = \left[ {\begin{aligned}0\\{ - 1}\\1\\{ - 1}\end{aligned}} \right]\]

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 1-6, the given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

5. \(\left( {\begin{aligned}{{}{}}1\\{ - 4}\\0\\1\end{aligned}} \right),\left( {\begin{aligned}{{}{}}7\\{ - 7}\\{ - 4}\\1\end{aligned}} \right)\)

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

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

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\[W\].

9.\[y = \left[ {\begin{aligned}4\\3\\3\\{ - 1}\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}1\\1\\0\\1\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 1}\\3\\1\\{ - 2}\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 1}\\0\\1\\1\end{aligned}} \right]\]
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