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Chapter 8: Q20E (page 437)
Question: Find an example to show that the convexity of S necessary in Exercise 19.
For \(c = 1\), \(d = 1\) and \(S = \left\{ {0,1} \right\} \subset \mathbb{R}\), the identity \(cS + dS = \left( {c + d} \right)S\) is not true as S is not convex.
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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 \(B\).
Explain why any set of five or more points in \({\mathbb{R}^3}\) must be affinely dependent.
Let\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)be an affinely dependent set of points in\({\mathbb{R}^{\bf{n}}}\)and let\(f:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\)be a linear transformation. Show that\(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\)is affinely dependent in\({\mathbb{R}^{\bf{m}}}\).
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: 25. Let \(p = \left( \begin{array}{l}1\\1\end{array} \right)\). Find a hyperplane \(\left( {f:d} \right)\) that strictly separates \(B\left( {0,3} \right)\) and \(B\left( {p,1} \right)\). (Hint: After finding \(f\), show that the point \(v = \left( {1 - .75} \right)0 + .75p\) is neither in \(B\left( {0,3} \right)\) nor in \(B\left( {p,1} \right)\)).
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