Chapter 8: Q18E (page 437)
In Exercises 17–20, prove the given statement about subsets \(A\) and \(B\) of \({R^n}\) . A proof for an exercise may use results of earlier exercises.
18. If \(A \subset B\), then \({\rm{conv}}\,A \subset {\rm{conv}}\,B\).
It is shown that\({\rm{conv}}\,A \subset {\rm{conv}}\,B\).
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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.
2.\(\left( {\begin{aligned}{{}}2\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}5\\4\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 3}\\{ - 2}\end{aligned}} \right)\)
Question: 26. Let \({\rm{q}} = \left( \begin{array}{l}2\\3\end{array} \right)\), \({\rm{p}} = \left( \begin{array}{l}6\\1\end{array} \right)\). Find a hyperplane \(\left( {f:d} \right)\) that strictly separates \(B\left( {{\rm{q}},3} \right)\) and \(B\left( {{\rm{p}},1} \right)\).
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.
1.\(\left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\0\end{aligned}} \right)\)
Let\(T\)be a tetrahedron in “standard” position, with three edges along the three positive coordinate axes in\({\mathbb{R}^3}\), and suppose the vertices are\(a{{\bf{e}}_1}\),\(b{{\bf{e}}_2}\),\(c{{\bf{e}}_{\bf{3}}}\), and 0, where\(\left[ {\begin{array}{*{20}{c}}{{{\bf{e}}_1}}&{{{\bf{e}}_2}}&{{{\bf{e}}_3}}\end{array}} \right] = {I_3}\). Find formulas for the barycentric coordinates of an arbitrary point\({\bf{p}}\)in\({\mathbb{R}^3}\).
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).
21. If \(A \subset B\), then B is affine, then \({\mathop{\rm aff}\nolimits} A \subset B\).
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