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Chapter 8: Q3E (page 437)

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

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{3}}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{4}}\\{ - {\bf{2}}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{6}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{{\bf{17}}}\\{\bf{1}}\\{\bf{5}}\end{aligned}} \right)\)

Short Answer

Expert verified

The affine combination is \({\bf{y}} = - 3{{\bf{v}}_1} + 2{{\bf{v}}_2} + 2{{\bf{v}}_3}\).

Step by step solution

01

Find the translated point

Write the translated points as shown below:

\({{\bf{v}}_2} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}3\\3\\{ - 3}\end{aligned}} \right)\)

\({{\bf{v}}_3} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}7\\{ - 3}\\5\end{aligned}} \right)\)

\({\bf{y}} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}{20}\\0\\4\end{aligned}} \right)\)

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

\(\begin{aligned}{c}{\bf{y}} - {{\bf{v}}_1} = {c_2}\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + {c_3}\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\\\left( {\begin{aligned}{*{20}{c}}{20}\\0\\4\end{aligned}} \right) = {c_2}\left( {\begin{aligned}{*{20}{c}}3\\3\\{ - 3}\end{aligned}} \right) + {c_3}\left( {\begin{aligned}{*{20}{c}}7\\{ - 3}\\5\end{aligned}} \right)\end{aligned}\)

02

Write the augmented matrix

The augmented matrix can be written as shown below:

\(M = \left( {\begin{aligned}{*{20}{c}}3&7&{20}\\3&{ - 3}&0\\{ - 3}&5&4\end{aligned}} \right)\)

Row reduce the augmented matrix as shown below:

\(\begin{aligned}{c}M = \left( {\begin{aligned}{*{20}{c}}3&7&{20}\\3&{ - 3}&0\\{ - 3}&5&4\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}3&7&{20}\\0&{ - 10}&{ - 20}\\0&{12}&{24}\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{aligned}{l}{R_2} \to {R_2} - {R_1}\\{R_3} \to {R_3} + {R_1}\end{aligned} \right)\\ = \left( {\begin{aligned}{*{20}{c}}3&7&{20}\\0&1&2\\0&1&2\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{aligned}{l}{R_2} \to - \frac{1}{{10}}{R_2}\\{R_3} \to \frac{1}{{12}}{R_3}\end{aligned} \right)\\ = \left( {\begin{aligned}{*{20}{c}}3&0&6\\0&1&2\\0&0&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{aligned}{l}{R_3} \to {R_3} - {R_2}\\{R_1} \to {R_1} - 7{R_2}\end{aligned} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&0&2\\0&1&2\\0&0&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{R_1} \to \frac{1}{3}{R_1}} \right)\end{aligned}\)

03

Write the system of equations

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

\({c_2} = 2\)

And,

\({c_3} = 2\)

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

\(\begin{aligned}{c}{\bf{y}} - {{\bf{v}}_1} = 2\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + 2\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\\{\bf{y}} = {{\bf{v}}_1} + 2{{\bf{v}}_2} - 2{{\bf{v}}_1} + 2{{\bf{v}}_3} - 2{{\bf{v}}_1}\\{\bf{y}} = - 3{{\bf{v}}_1} + 2{{\bf{v}}_2} + 2{{\bf{v}}_3}\end{aligned}\)

So, the vector \({\bf{y}}\) is \( - 3{{\bf{v}}_1} + 2{{\bf{v}}_2} + 2{{\bf{v}}_3}\).

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Most popular questions from this chapter

Questions: Let \({F_{\bf{1}}}\) and \({F_{\bf{2}}}\) be 4-dimensional flats in \({\mathbb{R}^{\bf{6}}}\), and suppose that \({F_{\bf{1}}} \cap {F_{\bf{2}}} \ne \phi \). What are the possible dimension of \({F_{\bf{1}}} \cap {F_{\bf{2}}}\)?

Question 1: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\3\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right)\) in \({\mathbb{R}^2}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a.\(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 3{x_1} + {x_2}\)

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{5}}\end{array}} \right]\),\({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{3}}\end{array}} \right]\),\({p_1} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{5}}\end{array}} \right]\),\({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\end{array}} \right]\),\({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\end{array}} \right]\),\({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{0}}\end{array}} \right]\),\({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\),\({p_{\bf{6}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\),\({p_{\bf{7}}} = \left[ {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{4}}\end{array}} \right]\)and let\(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of\({p_1}\),\({p_{\bf{2}}}\), and\({p_{\bf{3}}}\)with respect to S.
  3. On graph paper, sketch the triangle\(T\)with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\), extend the sides as in Figure 8, and plot the points\({p_{\bf{4}}}\),\({p_{\bf{5}}}\),\({p_{\bf{6}}}\), and\({p_{\bf{7}}}\). Without calculating the actual values, determine the signs of the barycentric coordinates of points\({p_{\bf{4}}}\),\({p_{\bf{5}}}\),\({p_{\bf{6}}}\), and\({p_{\bf{7}}}\).

Question: 18. Choose a set \(S\) of four points in \({\mathbb{R}^3}\) such that aff \(S\) is the plane \(2{x_1} + {x_2} - 3{x_3} = 12\). Justify your work.

Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{0}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).
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