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Chapter 8: Q22E (page 437)

Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).

Short Answer

Expert verified
  1. The given statement is False.
  2. The given statement is False.
  3. The given statement is True.
  4. The given statement is False.

Step by step solution

01

Use the definition of linear functional

For each scalar \(d\) in \(\mathbb{R}\), the symbol \(\left[ {f:d} \right]\) denotes the set of all \(x\) in \({\mathbb{R}^n}\) at which the value of \(f\) is \(d\).

So, the statement in (a) is true.

02

 Use theorem 11

According to theorem 11, if \(H\) is a hyperplane, there exists a nonzero vector \(n\) and a real number\(d\)such that \(H = \left\{ {x:n \cdot x = d} \right\}\).

So, the statement in (b) is false.

03

 Use theorem 12

According to theorem 12, the sets \(A\) and \(B\) must be nonempty and convex for a hyperplane \(H\) that strictly separates \(A\) and \(B\).

So, the statement in (c) is false.

04

 Use the concept of theorem 13

If no hyperplane \(H\) strictly separates \(A\) and \(B\), it implies that their convex hulls intersect. It might be that some other hyperplane not parallel to H would strictly separate them.

So, the statement in (d) is false.

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Most popular questions from this chapter

Explain why any set of five or more points in \({\mathbb{R}^3}\) must be affinely dependent.

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.

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Question: Let \({{\bf{a}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{1}}}\\{\bf{5}}\end{array}} \right)\), \({{\bf{a}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right)\), \({{\bf{a}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{6}}\\{\bf{0}}\end{array}} \right)\), \({{\bf{b}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{5}}\\{ - {\bf{1}}}\end{array}} \right)\), \({{\bf{b}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{ - {\bf{2}}}\end{array}} \right)\),\({{\bf{b}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{2}}\\{\bf{1}}\end{array}} \right)\) and \({\bf{n}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\), and let \(A = \left\{ {{{\bf{a}}_{\bf{1}}},{{\bf{a}}_{\bf{2}}},{{\bf{a}}_{\bf{3}}}} \right\}\) and \(B = \left\{ {{{\bf{b}}_{\bf{1}}},{{\bf{b}}_{\bf{2}}},{{\bf{b}}_{\bf{3}}}} \right\}\). Find a hyperplane H with normal n that separates A and B. Is there a hyperplane parallel to H that strictly separates A and B?

Question: In Exercise 8, let Hbe the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{2}}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{7}}\\{ - {\bf{4}}}\\{\bf{4}}\end{array}} \right)\)
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