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Chapter 8: Q24E (page 437)
Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in the statement of Exercise 23.
It is proved that \({\mathop{\rm aff}\nolimits} \left( {A \cup B} \right) \ne {\mathop{\rm aff}\nolimits} A \cup {\mathop{\rm aff}\nolimits} B\).
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Question: Let \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{3}}\\{\bf{0}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{1}}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\)
\({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{\bf{5}}\\{\bf{3}}\end{array}} \right)\) b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{9}}}\\{{\bf{10}}}\\{\bf{9}}\\{ - {\bf{13}}}\end{array}} \right)\) c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{2}}\\{\bf{8}}\\{\bf{5}}\end{array}} \right)\)
and \(S = \left\{ {{{\bf{v}}_1},\,\,{{\bf{v}}_2},\,{{\bf{v}}_3}} \right\}\). It can be shown that S is linearly independent.
a. Is \({{\bf{p}}_{\bf{1}}}\) is span S? Is \({{\bf{p}}_{\bf{1}}}\) is \({\bf{aff}}\,S\)?
b. Is \({{\bf{p}}_{\bf{2}}}\) is span S? Is \({{\bf{p}}_{\bf{2}}}\) is \({\bf{aff}}\,S\)?
c. Is \({{\bf{p}}_{\bf{3}}}\) is span S? Is \({{\bf{p}}_{\bf{3}}}\) is \({\bf{aff}}\,S\)?
Question: Let \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({\bf{n}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{5}}\\{ - {\bf{1}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{0}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and let H be the hyperplane in\({\mathbb{R}^{\bf{4}}}\) with normal n and passing through p. Which of the points \({{\bf{v}}_{\bf{1}}}\), \({{\bf{v}}_{\bf{2}}}\), and \({{\bf{v}}_{\bf{3}}}\) are on the same side of H as the origin, and which are not?
Question: 12. In Exercises 11 and 12, mark each statement True or False. Justify each answer.
a. If \(S = \left\{ {\bf{x}} \right\}\), then \({\rm{aff}}\,S\) is the empty set.
b. A set is affine if and only if it contains its affine hull.
c. A flat of dimension 1 is called a line.
d. A flat of dimension 2 is called a hyper plane.
e. A flat through the origin is a subspace.
Question: In Exercise 7, 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 \(H = \left( {f:d} \right)\).
7. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{4}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{ - {\bf{2}}}\\{\bf{5}}\end{array}} \right)\)
Question: 29. Prove that the open ball \(B\left( {{\rm{p}},\delta } \right) = \left\{ {{\rm{x:}}\left\| {{\rm{x - p}}} \right\| < \delta } \right\}\)is a convex set. (Hint: Use the Triangle Inequality).
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