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Chapter 8: Q4E (page 437)

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

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{ - {\bf{6}}}\\{\bf{7}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}\\{\bf{3}}\\{\bf{1}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{3}}}\\{\bf{4}}\\{ - {\bf{4}}}\end{aligned}} \right)\)

Short Answer

Expert verified

The affine combination is \({\bf{y}} = \frac{1}{5}{{\bf{v}}_1} - \frac{2}{5}{{\bf{v}}_2} + \frac{6}{5}{{\bf{v}}_3}\).

Step by step solution

01

Find the translated point

Write the translated points as shown below:

\({{\bf{v}}_2} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}1\\8\\{ - 7}\end{aligned}} \right)\)

\({{\bf{v}}_3} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}3\\1\\1\end{aligned}} \right)\)

\({\bf{y}} - {{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}{ - 4}\\2\\4\end{aligned}} \right)\)

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

\(\begin{aligned}{c}{\bf{y}} - {{\bf{v}}_1} = {c_2}\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + {c_3}\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\\\left( {\begin{aligned}{*{20}{c}}{ - 4}\\2\\4\end{aligned}} \right) = {c_2}\left( {\begin{aligned}{*{20}{c}}1\\{ - 8}\\7\end{aligned}} \right) + {c_3}\left( {\begin{aligned}{*{20}{c}}3\\1\\1\end{aligned}} \right)\end{aligned}\)

02

Write the augmented matrix

The augmented matrix can be written as shown below:

\(M = \left( {\begin{aligned}{*{20}{c}}1&3&{ - 4}\\{ - 8}&1&2\\7&1&{ - 4}\end{aligned}} \right)\)

Row reduce the augmented matrix as shown below:

\(\begin{aligned}{c}M = \left( {\begin{aligned}{*{20}{c}}1&3&{ - 4}\\{ - 8}&1&2\\7&1&{ - 4}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&3&{ - 4}\\0&{25}&{ - 30}\\0&{ - 20}&{24}\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{aligned}{l}{R_2} \to {R_2} + 8{R_1}\\{R_3} \to {R_3} - 7{R_1}\end{aligned} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&3&{ - 4}\\0&5&{ - 6}\\0&{ - 10}&{12}\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{aligned}{l}{R_2} \to \frac{1}{5}{R_2}\\{R_3} \to \frac{1}{2}{R_3}\end{aligned} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&3&{ - 4}\\0&1&{ - \frac{6}{5}}\\0&0&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \begin{aligned}{l}{R_3} \to {R_3} + 2{R_2}\\{R_2} \to \frac{1}{5}{R_2}\end{aligned} \right)\\ = \left( {\begin{aligned}{*{20}{c}}1&0&{ - \frac{2}{5}}\\0&1&{\frac{6}{5}}\\0&0&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{R_1} \to {R_1} - 3{R_2}} \right)\end{aligned}\)

03

Write the system of equations

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

\({c_2} = - \frac{2}{5}\)

And,

\({c_3} = \frac{6}{5}\)

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

\(\begin{aligned}{c}{\bf{y}} - {{\bf{v}}_1} = - \frac{2}{5}\left( {{{\bf{v}}_2} - {{\bf{v}}_1}} \right) + \frac{6}{5}\left( {{{\bf{v}}_3} - {{\bf{v}}_1}} \right)\\{\bf{y}} = {{\bf{v}}_1} - \frac{2}{5}{{\bf{v}}_2} + \frac{2}{5}{{\bf{v}}_1} + \frac{6}{5}{{\bf{v}}_3} - \frac{6}{5}{{\bf{v}}_1}\\{\bf{y}} = \frac{1}{5}{{\bf{v}}_1} - \frac{2}{5}{{\bf{v}}_2} + \frac{6}{5}{{\bf{v}}_3}\end{aligned}\)

So, the vector \({\bf{y}}\) is \(\frac{1}{5}{{\bf{v}}_1} - \frac{2}{5}{{\bf{v}}_2} + \frac{6}{5}{{\bf{v}}_3}\).

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Most popular questions from this chapter

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{5}}\\{\bf{0}}&{\bf{2}}&{\bf{4}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\). (Hint: How is H is related to Nul B?see section 6.1.)

Question: 30. Prove that the convex hull of a bounded set is bounded.

Question: In Exercise 8, let Hbe the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{2}}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{7}}\\{ - {\bf{4}}}\\{\bf{4}}\end{array}} \right)\)

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

21. If \(A \subset B\), then B is affine, then \({\mathop{\rm aff}\nolimits} A \subset B\).

In Exercises 21鈥�24, a, b, and c are noncollinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \)and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

24. Take q on the line segment from b to c and consider the line through q and a, which may be written as\(p = \left( {1 - x} \right)q + xa\)for all real x. Show that, for each x,\(det\left[ {\begin{array}{*{20}{c}}{\widetilde p}&{\widetilde b}&{\widetilde c}\end{array}} \right] = x \cdot det\left[ {\begin{array}{*{20}{c}}{\widetilde a}&{\widetilde b}&{\widetilde c}\end{array}} \right]\). From this and earlier work, conclude that the parameter x is the first barycentric coordinate of p. However, by construction, the parameter x also determines the relative distance between p and q along the segment from q to a. (When x = 1, p = a.) When this fact is applied to Example 5, it shows that the colors at vertex a and the point q are smoothly interpolated as p moves along the line between a and q.
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