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Chapter 8: Q8.1-14E (page 437)

Question: 14. Show that if \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\), then aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).

Short Answer

Expert verified

It is shown that aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).

Step by step solution

01

Describe the given statement

It is given that \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is in \({\mathbb{R}^3}\) , and it is also known that span of \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) is in the plane \({\mathbb{R}^3}\). Let denote it by \(W\).

02

 Use linear transformation theory

If \(W = {\rm{span}}\left\{ {{{\bf{v}}_{\rm{2}}} - {{\bf{v}}_{\rm{1}}},{{\bf{v}}_{\rm{3}}} - {{\bf{v}}_{\rm{1}}}} \right\}\), then the plane \(W\) which consists \({{\rm{v}}_{\rm{1}}}\) is parallel to \(W + {{\rm{v}}_{\rm{1}}}\). As \({{\rm{v}}_{\rm{1}}}\), \({{\bf{v}}_{\bf{2}}}\) and \({{\rm{v}}_{\rm{3}}}\) are the linear combination of each other, \(W + {{\bf{v}}_{\bf{1}}}\) must contain \({{\rm{v}}_{\rm{2}}}\), \({{\bf{v}}_{\bf{3}}}\) and so on.

03

 Use Theorem 1

According to theorem 1, \(y - {{\bf{v}}_{\bf{1}}}\) are the combination of points \({{\bf{v}}_{\bf{2}}} - {{\bf{v}}_{\bf{1}}},...,{{\bf{v}}_{\bf{p}}} - {{\bf{v}}_{\bf{1}}}\), if the point \(y\) is an affine combination of \({{\bf{v}}_{\bf{1}}}{\bf{,}}...,{{\bf{v}}_{\bf{p}}}\) in \({\mathbb{R}^n}\).

So, the plane \(W + {{\bf{v}}_{\bf{1}}}\) is an affine combination of \({{\bf{v}}_{\bf{1}}}{\bf{,}}{\rm{ }}{{\bf{v}}_{\bf{2}}}{\bf{,}}{\rm{ }}{{\bf{v}}_{\bf{3}}}\) as \(W + {{\bf{v}}_{\bf{1}}}\) are combination of points \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) in \({\mathbb{R}^3}\).

Thus, it is shown that aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).

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Most popular questions from this chapter

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

In Exercises 7 and 8, find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.

7. \(\left( {\begin{array}{{}}1\\{ - 1}\\2\\1\end{array}} \right),\left( {\begin{array}{{}}2\\1\\0\\1\end{array}} \right),\left( {\begin{array}{{}}1\\2\\{ - 2}\\0\end{array}} \right)\), \({\mathop{\rm p}\nolimits} = \left( {\begin{array}{{}}5\\4\\{ - 2}\\2\end{array}} \right)\)

Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{0}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).

Question:28. Give an example of a compact set\(A\)and a closed set\(B\)in\({\mathbb{R}^2}\)such that\(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) = \emptyset \)but\(A\)and\(B\)cannot be strictly separated by a hyperplane.

Question: Let \({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{ - {\bf{1}}}\\{\bf{3}}\end{array}} \right)\), \({{\bf{n}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{4}}\\{\bf{2}}\end{array}} \right)\), and \({{\bf{n}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\\{\bf{1}}\\{\bf{5}}\end{array}} \right)\), let \({H_{\bf{1}}}\) be the hyperplane in \({\mathbb{R}^{\bf{4}}}\) through \({{\bf{p}}_{\bf{1}}}\) with normal \({{\bf{n}}_{\bf{1}}}\), and let \({H_{\bf{2}}}\) be the hyperplane through \({{\bf{p}}_{\bf{2}}}\) with normal \({{\bf{n}}_{\bf{2}}}\). Give an explicit description of \({H_{\bf{1}}} \cap {H_{\bf{2}}}\). (Hint: Find a point p in \({H_{\bf{1}}} \cap {H_{\bf{2}}}\) and two linearly independent vectors \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) that span a subspace parallel to the 2-dimensional flat \({H_{\bf{1}}} \cap {H_{\bf{2}}}\).)
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