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Chapter 8: Q5E (page 437)

Question: In Exercises 5-8, find the minimal representation of the polytope defined by the inequalities \(A{\bf{x}} \le {\bf{b}}\) and \({\bf{x}} \ge {\bf{0}}\).

5. \(A = \left( {\begin{array}{*{20}{c}}1&2\\3&1\end{array}} \right),{\rm{ }}{\bf{b}} = \left( {\begin{array}{*{20}{c}}{{\bf{10}}}\\{{\bf{15}}}\end{array}} \right)\)

Short Answer

Expert verified

The minimal representation of the polytope \(P\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}5\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}4\\3\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\5\end{array}} \right)} \right\}\).

Step by step solution

01

The two inequalities in \(A{\bf{x}} \le {\bf{b}}\)

The matrix inequalities \(A{\bf{x}} \le {\bf{b}}\) yield the following system of inequalities:

  1. \({{\mathop{\rm x}\nolimits} _1} + 2{{\mathop{\rm x}\nolimits} _2} \le 10\)
  2. \(3{{\mathop{\rm x}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2} \le 15\)
02

Determine the \({{\bf{x}}_1}\)-intercept and \({{\bf{x}}_2}\)-intercept of the two lines

The condition \({\mathop{\rm x}\nolimits} \ge 0\) places polytope \(P\) in the first quadrant of the plane. One vertex is \(\left( {0,0} \right)\).

The \({{\mathop{\rm x}\nolimits} _1}\)-intercepts \(\left( {{\mathop{\rm If}\nolimits} {\rm{ }}{{\mathop{\rm x}\nolimits} _2} = 0} \right)\) of the two lines are 10 and 5, so \(\left( {5,0} \right)\) is a vertex.

The \({{\mathop{\rm x}\nolimits} _2}\)-intercepts\(\left( {{\mathop{\rm If}\nolimits} {\rm{ }}{{\mathop{\rm x}\nolimits} _1} = 0} \right)\) of the two lines are 5 and 15, then \(\left( {0,5} \right)\) is a vertex.

03

Determine the intersection point corresponds to inequalities

The intersection of (a) is at \({{\mathop{\rm P}\nolimits} _{\mathop{\rm a}\nolimits} } = \left( {4,3} \right)\). Testing \({{\mathop{\rm P}\nolimits} _a}\) in (b) gives \(3\left( 4 \right) + 3 = 15\), so \({{\mathop{\rm P}\nolimits} _a}\) is in \({\mathop{\rm P}\nolimits} \).

04

Determine the minimal representation of the polytope

The four vertices of the polytope are \(\left( {0,0} \right),\left( {5,0} \right)\left( {4,3} \right),\,\,{\mathop{\rm and}\nolimits} \,\,\left( {0,5} \right)\).

Thus, the minimal representation of the polytope \(P\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}5\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}4\\3\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\5\end{array}} \right)} \right\}\).

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Most popular questions from this chapter

Question: 13. Suppose \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\). Show that Span \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) is a plane in \({\mathbb{R}^3}\). (Hint: What can you say about \({\rm{u}}\) and \({\rm{v}}\)when Span \(\left\{ {{\rm{u,v}}} \right\}\) is a plane?)

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 \({\bf{1}} \times {\bf{4}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{2}}}\end{array}} \right)\) and let \(b = {\bf{5}}\). Let \(H = \left\{ {{\bf{x}}\,\,{\rm{in}}\,{\mathbb{R}^{\bf{4}}}:A{\bf{x}} = {\bf{b}}} \right\}\).

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

10.a. If \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely dependent set in \({\mathbb{R}^n}\), then the set \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) of homogeneous forms may be linearly independent.

b. If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) and \({{\mathop{\rm v}\nolimits} _4}\) are in \({\mathbb{R}^3}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _4} - {{\mathop{\rm v}\nolimits} _1}} \right\}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) is affinely independent.

c. Given \(S = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}} \right\}\) in \({\mathbb{R}^n}\), each \({\bf{p}}\) in\({\mathop{\rm aff}\nolimits} S\) has a unique representation as an affine combination of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}\).

d. When color information is specified at each vertex \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) of a triangle in \({\mathbb{R}^3}\), then the color may be interpolated at a point p in \({\mathop{\rm aff}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) using the barycentric coordinates of p.

e. If T is a triangle in \({\mathbb{R}^2}\) and if a point p is on edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.

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

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