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Chapter 8: Q2E (page 437)

Question 2: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}0\\{ - 1}\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\1\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right)\) in \({\mathbb{R}^{\bf{2}}}\), 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. \(f\left( {{x_1},{x_2}} \right) = - 2{x_1} + {x_2}\)

Short Answer

Expert verified
  1. On the set conv\(\left\{ {{{\bf{p}}_2},{{\bf{p}}_3}} \right\}\) is the point in \(S\) at \(m = 3\).
  2. On the set \({\rm{conv}}\left\{ {{{\bf{p}}_1},{{\bf{p}}_2}} \right\}\) is the point in \(S\) at \(m = 1\).
  3. \({{\mathop{\rm p}\nolimits} _3}\) is the point in \(S\) at \(m = 0\).

Step by step solution

01

The maximum and minimum is attained at an extreme point

Theorem 16states that consider f as a linear functional defined on a nonempty compact convex set \(S\).

Then, there are extreme points \(\widehat {\mathop{\rm v}\nolimits} \) and \(\widehat {\mathop{\rm w}\nolimits} \) of \(S\) such that \(f\left( {\widehat {\mathop{\rm v}\nolimits} } \right) = \mathop {\max }\limits_{{\mathop{\rm v}\nolimits} \in S} f\left( {\mathop{\rm v}\nolimits} \right)\) and \(f\left( {\widehat {\mathop{\rm w}\nolimits} } \right) = \mathop {\min }\limits_{{\mathop{\rm v}\nolimits} \in S} f\left( {\mathop{\rm v}\nolimits} \right)\).

02

Determine the maximum value \(m\) of \(f\)

According to theorem 16, the maximum value is attained at one of the extreme points of \(S\).

Evaluate \(f\) at the extreme point and select the largest value to find \(m\) as shown below:

  1. \({f_1}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = - 1\), \({f_1}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = 3\), \({f_1}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = 3\), therefore, \({m_1} = 3\). Graph the line \({f_1}\left( {{x_1},{x_2}} \right) = {m_1}\) which means that \({x_1} + {x_2} = 3\), and \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _2}\) is the only point in \(S\) at which \({f_1}\left( {\mathop{\rm x}\nolimits} \right) = 3\).


b. \({f_2}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = 1\), \({f_2}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = 1\), \({f_2}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = - 1\), therefore, \({m_2} = 1\). Graph the line \({f_2}\left( {{x_1},{x_2}} \right) = {m_2}\) which means that \({x_1} - {x_2} = 1\), and \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _1}\) is the only point in \(S\) at which \({f_2}\left( {\mathop{\rm x}\nolimits} \right) = 1\).


c. \({f_3}\left( {{{\mathop{\rm p}\nolimits} _1}} \right) = - 1\), \({f_3}\left( {{{\mathop{\rm p}\nolimits} _2}} \right) = - 3\), \({f_3}\left( {{{\mathop{\rm p}\nolimits} _3}} \right) = 0\), therefore, \({m_3} = 0\). Graph the line \({f_3}\left( {{x_1},{x_2}} \right) = {m_3}\) which means that \( - 2{x_1} + {x_2} = 0\), and \({\mathop{\rm x}\nolimits} = {{\mathop{\rm p}\nolimits} _3}\) is the only point in \(S\) at which \({f_3}\left( {\mathop{\rm x}\nolimits} \right) = 0\).


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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 \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{4}}&{ - {\bf{5}}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{8}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\).

Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in the statement of Exercise 25.

Let \({\bf{x}}\left( t \right)\) be a cubic B茅zier curve determined by points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\), \({{\bf{p}}_2}\), and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\). Determine how \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have \({\bf{x}}'\left( 1 \right) = 0\)?

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_1}\) as the origin of the coordinate system.]

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.

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9.

a. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _1} - {{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _2}} \right\}\) is linearly dependent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent. (Read this carefully.)

b. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set of homogeneous forms \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent.

c. A finite set of points \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is affinely dependent if there exist real numbers \({c_1},...,{c_k}\) , not all zero, such that \({c_1} + ... + {c_k} = 1\) and \({c_1}{{\mathop{\rm v}\nolimits} _1} + ... + {c_k}{{\mathop{\rm v}\nolimits} _k} = 0\).

d. If \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely independent set in \({\mathbb{R}^n}\) and if p in \({\mathbb{R}^n}\) has a negative barycentric coordinate determined by S, then p is not in \({\mathop{\rm aff}\nolimits} S\).

e.

If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},a,\) and \(b\) are in \({\mathbb{R}^3}\) and if ray \({\mathop{\rm a}\nolimits} + t{\mathop{\rm b}\nolimits} \) for \(t \ge 0\) intersects the triangle with vertices \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\) then the barycentric coordinates of the intersection points are all nonnegative.
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