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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: 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 \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{4}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).

Let \(W = \left\{ {{{\bf{v}}_1},......,{{\bf{v}}_p}} \right\}\). Show that if \({\bf{x}}\) is orthogonal to each \({{\bf{v}}_j}\), for \(1 \le j \le p\), then \({\bf{x}}\) is orthogonal to every vector in \(W\).

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 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).

25. \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

6.\(\left( {\begin{aligned}{{}}1\\3\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\{ - 1}\\{ - 2}\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\5\\2\end{aligned}} \right),\left( {\begin{aligned}{{}}3\\5\\0\end{aligned}} \right)\)
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