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Chapter 8: Q17E (page 437)

Question: In Exercise 16 and 17, mark each statement True or False. Justify each answer.

a. A cube in \({\mathbb{R}^{\bf{3}}}\) has exactly five facets.

b. A point p is an extreme point of a polytope P if and only if p is a vertex of P.

c. If S is a nonempty compact convex set and a linear functional attains its maximum at a point p, then p is an extreme point of S.

d. A 2-dimensional polytope always has the same number of vertices and edges.

Short Answer

Expert verified

a. The given statement is False.

b. The given statement is True.

c. The given statement is False.

d. The given statement is True.

Step by step solution

01

Check for the statement (a)

A cube in \({\mathbb{R}^3}\) has exactly 6 facets.

So, the given statement (a) is False.

02

Check for the statement (b)

According to theorem-14, a point p is an extreme point of a polytope P if and only if p is a vertex of P.

So, the given statement (b) is True.

03

Check for the statement (c)

Recall Theorem 16: the maximum can also be obtained at other points even though the maximum is always at some extreme point.

So, the given statement is False.

04

Check for the statement (d)

By the Euler formula \(n = 2\), A 2-dimensional polytope always has the same number of vertices and edges.

So, the given statement (d) is True.

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Most popular questions from this chapter

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\3\\{ - 6}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{7}}\\3\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{9}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{9}}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{1.4}\\{{\bf{1}}.{\bf{5}}}\\{ - {\bf{3}}.{\bf{1}}}\end{array}} \right]\), and \({\bf{x}}\left( t \right) = {\bf{a}} + t{\bf{b}}\)for \(t \ge {\bf{0}}\).Find the point where the ray\({\bf{x}}\left( t \right)\)intersects the plane that contains the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). Is this point inside the triangle?

Question: 14. Show that if \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\), then aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).

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 A be the \({\bf{1}} \times {\bf{5}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{2}}&{\bf{5}}&{ - {\bf{3}}}&{\bf{0}}&{\bf{6}}\end{array}} \right)\). Note that \({\bf{Nul}}\,\,A\) is in \({\mathbb{R}^{\bf{5}}}\). Let \(H = {\bf{Nul}}\,\,A\).

In Exercises 5 and 6, let \({{\bf{b}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right)\), \({{\bf{b}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{ - {\bf{2}}}\end{aligned}} \right)\), and \({{\bf{b}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{ - {\bf{5}}}\\{\bf{1}}\end{aligned}} \right)\) and \(S = \left\{ {{{\bf{b}}_{\bf{1}}},\,{{\bf{b}}_{\bf{2}}},\,{{\bf{b}}_{\bf{3}}}} \right\}\). Note that S is an orthogonal basis of \({\mathbb{R}^{\bf{3}}}\). Write each of the given points as an affine combination of the points in the set S, if possible. (Hint: Use Theorem 5 in section 6.2 instead of row reduction to find the weights.)

a. \({{\bf{p}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{8}}\\{\bf{4}}\end{aligned}} \right)\)

b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{6}}\\{ - {\bf{3}}}\\{\bf{3}}\end{aligned}} \right)\)

c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{ - {\bf{1}}}\\{ - {\bf{5}}}\end{aligned}} \right)\)

Use only the definition of affine dependence to show that anindexed set \(\left\{ {{v_1},{v_2}} \right\}\) in \({\mathbb{R}^{\bf{n}}}\) is affinely dependent if and only if \({v_1} = {v_2}\).
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