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Chapter 8: Q8.1-18E (page 437)

Question: 18. Choose a set \(S\) of four points in \({\mathbb{R}^3}\) such that aff \(S\) is the plane \(2{x_1} + {x_2} - 3{x_3} = 12\). Justify your work.

Short Answer

Expert verified

The set is \(S = \left\{ {\left( \begin{array}{l}2\\0\\0\end{array} \right),\left( \begin{array}{l}\,\,1\\12\\\,\,0\end{array} \right),\left( \begin{array}{l}\,\,0\\\,\,0\\ - 4\end{array} \right),\left( \begin{array}{l}\,\,4\\\,\,3\\ - 1\end{array} \right)} \right\}\).

Step by step solution

01

Describe the given statement

The set of four vectors that lie along the plane \(2{x_1} + {x_2} - 3{x_3} = 12\) cannot be collinear.

The set of vectors that is not collinear cannot have a line as their affine hull.

02

 Draw a conclusion

One of the possible sets of three vectors discussed above is \(S = \left\{ {\left( \begin{array}{l}2\\0\\0\end{array} \right),\left( \begin{array}{l}\,\,1\\12\\\,\,0\end{array} \right),\left( \begin{array}{l}\,\,0\\\,\,0\\ - 4\end{array} \right),\left( \begin{array}{l}\,\,4\\\,\,3\\ - 1\end{array} \right)} \right\}\).

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Most popular questions from this chapter

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\({\mathbb{R}^3}\).

Question: 15. Let \(A\) be an \({\rm{m}} \times {\rm{n}}\) matrix and, given \({\rm{b}}\) in \({\mathbb{R}^m}\), show that the set \(S\) of all solutions of \(A{\rm{x}} = {\rm{b}}\) is an affine subset of \({\mathbb{R}^n}\).

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.

6.\(\left( {\begin{aligned}{{}}1\\3\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\{ - 1}\\{ - 2}\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\5\\2\end{aligned}} \right),\left( {\begin{aligned}{{}}3\\5\\0\end{aligned}} \right)\)

Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in the statement of Exercise 25.

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?
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