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Chapter 8: Q1E (page 437)

In \({\mathbb{R}^{\bf{2}}}\), let \(S = \left\{ {\left( {\begin{aligned}{{}{}}{\bf{0}}\\y\end{aligned}} \right):{\bf{0}} \le y < {\bf{1}}} \right\} \cup \left\{ {\left( {\begin{aligned}{{}{}}{\bf{2}}\\{\bf{0}}\end{aligned}} \right)} \right\}\). Describe (or sketch) the convex hull of S.

Short Answer

Expert verified

\({k_1}{v_1} + {k_2}{v_2} + ...... + {k_n}{v_n} + \left( 0 \right){v_{n + 1}}\)

Step by step solution

01

Step 1:Find constants for convex of the hull

Let \({c_1}\), \({c_2}\),鈥�., \({c_n}\)be constants(\({c_i} \ge 0,\,\forall i\)). Alsofor some constants\({k_1}\), \({k_2}\),鈥�.,\({k_n}\),

\({k_1} = \frac{{{c_1}}}{{{c_1} + {c_2} + .... + {c_n}}},\,\,{k_2} = \frac{{{c_2}}}{{{c_1} + {c_2} + .... + {c_n}}}\,,..............,{k_n} = \frac{{{c_n}}}{{{c_1} + {c_2} + .... + {c_n}}}\).

And

\({c_1} + {c_2} + ...... + {c_n} = 1\).

For the above constants,

\(\begin{aligned}{}{k_1} + {k_2} + .... + {k_n} &= \frac{{{c_1}}}{{{c_1} + {c_2} + ... + {c_n}}} + ..... + \frac{{{c_n}}}{{{c_1} + {c_2} + ... + {c_n}}}\\ &= \frac{{{c_1} + {c_2} + ..... + {c_n}}}{{{c_1} + {c_2} + ..... + {c_n}}}\\ &= 1\end{aligned}\)

02

Find the convex hull

The given vector sets are:

\({v_1} = \left( {\begin{aligned}{{}{}}0\\{{y_1}}\end{aligned}} \right)\), \({v_2} = \left( {\begin{aligned}{{}{}}0\\{{y_2}}\end{aligned}} \right)\),鈥︹�︹�︹��,\({v_n} = \left( {\begin{aligned}{{}{}}0\\{{y_n}}\end{aligned}} \right)\), \({v_{n + 1}} = \left( {\begin{aligned}{{}{}}2\\0\end{aligned}} \right)\)

Therefore,theconvex hullhas elements in it, which is given by the expression:

\({k_1}{v_1} + {k_2}{v_2} + ...... + {k_n}{v_n} + \left( 0 \right){v_{n + 1}}\)

So, the vector \({\bf{y}}\) is \(2{{\bf{v}}_1} - \frac{3}{2}{{\bf{v}}_2} + \frac{1}{2}{{\bf{v}}_3}\).

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Most popular questions from this chapter

Question: 19. Let \(S\) be an affine subset of \({\mathbb{R}^n}\) , suppose \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation, and let \(f\left( S \right)\) denote the set of images \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). Prove that \(f\left( S \right)\)is an affine subset of \({\mathbb{R}^m}\).

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: 20. Let \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\) is a linear transformation, and let \(T\) be an affine subset of \({\mathbb{R}^{\bf{m}}}\), and let \(S = \left\{ {{\bf{x}} \in {\mathbb{R}^n}\,:\,f\left( {\bf{x}} \right) \in T} \right\}\). Show that \(S\) is an affine subset of \({\mathbb{R}^m}\).

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 \({\bf{aff}}\,S\)?

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