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Chapter 8: Q20E (page 437)

Question:20.In Exercises 17–20, prove the given statement about subsets \(A\) and \(B\) of \({\mathbb{R}^n}\) . A proof for an exercise may use results of earlier exercises.

20. a. \({\rm{conv}}\,\left( {{\rm{A}} \cap {\rm{B}}} \right) \subset \left( {\left( {{\rm{conv}}\,A} \right)\,\, \cap \left( {{\rm{conv}}\,B} \right)} \right)\),

b. Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in part (a).

Short Answer

Expert verified
  1. It is shown that \({\rm{conv}}\,\left( {{\rm{A}} \cap {\rm{B}}} \right) \subset \left( {\left( {{\rm{conv}}\,A} \right)\,\, \cap \left( {{\rm{conv}}\,B} \right)} \right)\).
  2. A square with \(A\) be the two adjacent corners of the square, and B be the other two corners of the square.

Step by step solution

01

Step 1:Use the result of Exercise 18

The combinations of A must contain combinations of \(A\)and\(B\). So, \(\,\left( {A \cap B} \right) \subset A\) .

Similarly,a convex combination of \(A\) must containsevery convex combination of \(A\) and\(B\);that is, \({\rm{conv}}\,\left( {{\rm{A}} \cap \,B} \right) \subset {\rm{conv}}\,A\)or \({\rm{conv}}\,\left( {{\rm{A}} \cap \,B} \right) \subset {\rm{conv}}\,B\).

02

Step 2:Draw a conclusion

If \({\rm{conv}}\,\left( {{\rm{A}} \cap \,B} \right) \subset {\rm{conv}}\,A\) is true, then \({\rm{conv}}\,\left( {{\rm{A}} \cap {\rm{B}}} \right) \subset \left( {\left( {{\rm{conv}}\,A} \right)\,\, \cap \left( {{\rm{conv}}\,B} \right)} \right)\) must also be true as a convex combination of points ofAandconvex combination of points of \(B\)must contain some or all points of convex combinations of \(A\)and\(B\).

03

 Assume an example in \({\mathbb{R}^2}\) as a requirement

Consider a square and assume \(A\) be the two diagonally opposite corners of the square, whereas B be the other diagonally opposite corners of the square.

Then, \({\rm{conv}}\,A\,\, \cap {\rm{conv}}\,B\) represents the diagonals of the square and contains their intersection point, whereas \({\rm{conv}}\,\left( {A\,\, \cap \,B} \right)\)should be an empty set as \(A\,\, \cap \,B\) is empty.

So, \({\rm{conv}}\,\left( {{\rm{A}} \cap {\rm{B}}} \right) \subset \left( {\left( {{\rm{conv}}\,A} \right)\,\, \cap \left( {{\rm{conv}}\,B} \right)} \right)\)is followed.

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Most popular questions from this chapter

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 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, \(H = {\bf{Row}}\,B\).

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.

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.

The conditions for affine dependence are stronger than those for linear dependence, so an affinely dependent set is automatically linearly dependent. Also, a linearly independent set cannot be affinely dependent and therefore must be affinely independent. Construct two linearly dependent indexed sets\({S_{\bf{1}}}\)and\({S_{\bf{2}}}\)in\({\mathbb{R}^2}\)such that\({S_{\bf{1}}}\)is affinely dependent and\({S_{\bf{2}}}\)is affinely independent. In each case, the set should contain either one, two, or three nonzero points.
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