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

Question: Repeat Exercise 7 when

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{3}}\\{ - {\bf{2}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{6}}\\{ - {\bf{5}}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{0}}\\{{\bf{12}}}\\{ - {\bf{6}}}\end{array}} \right)\)

\({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{1}}}\\{{\bf{15}}}\\{ - {\bf{7}}}\end{array}} \right)\), \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{5}}}\\{\bf{3}}\\{ - {\bf{8}}}\\{\bf{6}}\end{array}} \right)\), and \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{6}}\\{ - {\bf{6}}}\\{ - {\bf{8}}}\end{array}} \right)\)

Short Answer

Expert verified

a. \({{\bf{p}}_1} \in {\rm{span}}\,S\), but \({{\bf{p}}_1} \notin \,{\rm{aff}}\,\,S\)

b. \({{\bf{p}}_2} \notin {\rm{span}}\,S\), but \({{\bf{p}}_2} \notin {\rm{aff}}\,\,S\)

c. \({{\bf{p}}_3} \in {\rm{span}}\,S\), but \({{\bf{p}}_3} \in {\rm{aff}}\,\,S\)

Step by step solution

01

Find the augmented matrix

Write the augmented matrix by using the points as shown below:

\(\begin{array}{c}M = \left( {\begin{array}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_2}}&{{{\bf{b}}_3}}&{{{\bf{p}}_1}}&{{{\bf{p}}_2}}&{{{\bf{p}}_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}1&2&3&4&{ - 5}&4\\0&1&0&{ - 1}&3&2\\3&6&{12}&{15}&{ - 8}&8\\{ - 2}&{ - 5}&{ - 6}&{ - 7}&6&5\end{array}} \right)\end{array}\)

02

Write the row reduced form of the augmented matrix

The augmented matrix can be written as,

\(\begin{array}{c}M = \left( {\begin{array}{*{20}{c}}1&2&3&4&{ - 5}&4\\0&1&0&{ - 1}&3&6\\0&0&3&3&7&{ - 9}\\0&{ - 1}&0&1&{ - 4}&{ - 6}\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_3} \to {R_3} - 3{R_1}\\{R_4} \to {R_4} + 2{R_1}\end{array} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&3&6&{ - 11}&{ - 11}\\0&1&0&{ - 1}&3&6\\0&0&3&3&7&{ - 9}\\0&0&0&0&{ - 1}&0\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_1} \to {R_1} - 2{R_2}\\{R_4} \to {R_4} + {R_2}\end{array} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&0&3&{ - 18}&{ - 2}\\0&1&0&{ - 1}&0&6\\0&0&3&3&7&{ - 9}\\0&0&0&0&{ - 1}&0\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_1} \to {R_1} - {R_3}\\{R_2} \to {R_2} + 3{R_4}\end{array} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&0&3&0&{ - 2}\\0&1&0&{ - 1}&0&6\\0&0&3&3&7&{ - 9}\\0&0&0&0&{ - 1}&0\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_1} \to {R_1} - 18{R_4}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&0&3&0&{ - 2}\\0&1&0&{ - 1}&0&6\\0&0&1&1&{\frac{7}{3}}&{ - 3}\\0&0&0&0&{ - 1}&0\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to \frac{1}{3}{R_3}} \right\}\end{array}\)

There are no zero rows in the augmented matrix. Hence the set is linearly independent.

03

Check for the affine combination of \({{\bf{p}}_{\bf{1}}}\)

Use the augmented matrix, \({{\bf{p}}_1}\) which can be expressed as shown below:

\({{\bf{p}}_1} = 3\left( {{{\bf{b}}_1}} \right) - 1\left( {{{\bf{b}}_2}} \right) + 1\left( {{{\bf{b}}_3}} \right)\)

The sum of coefficients is \(3 - 1 + 1 = 3 \ne 1\).

So, \({{\bf{p}}_1}\) is not an affine combination of point in S.

04

Check for the affine combination of \({{\bf{p}}_{\bf{2}}}\)

From the augmented matrix, it can be observed that, \({{\bf{p}}_2}\) can not be written as the linear combination of point of S.

\({{\bf{p}}_2}\)is not an affine combination of points in S.

05

Check for an affine combination of \({{\bf{p}}_{\bf{3}}}\)

Use the augmented matrix, \({{\bf{p}}_3}\) which can be expressed as shown below:

\({{\bf{p}}_3} = - 2\left( {{{\bf{b}}_1}} \right) + 6\left( {{{\bf{b}}_2}} \right) - 3\left( {{{\bf{b}}_3}} \right)\)

The sum of coefficients is \( - 2 + 6 - 3 = 1\).

So, \({{\bf{p}}_3}\) is an affine combination of point in S.

\({{\bf{p}}_3} = - 2\left( {{{\bf{b}}_1}} \right) + 6\left( {{{\bf{b}}_2}} \right) - 3\left( {{{\bf{b}}_3}} \right)\)

Hence, \({{\bf{p}}_3}\) is an affine combination of point in S.

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Most popular questions from this chapter

Question: 11. In Exercises 11 and 12, mark each statement True or False. Justify each answer.

11. a. The set of all affine combinations of points in a set \(S\) is called the affine hull of \(S\).

b. If \(\left\{ {{{\rm{b}}_{\rm{1}}}{\rm{,}}.......{{\rm{b}}_{\rm{2}}}} \right\}\) is a linearly independent subset of \({\mathbb{R}^n}\) and if \({\bf{p}}\) is a linear combination of \({{\rm{b}}_{\rm{1}}}.......{{\rm{b}}_{\rm{k}}}\), then \({\rm{p}}\) is an affine combination of \({{\rm{b}}_{\rm{1}}}.......{{\rm{b}}_{\rm{k}}}\).

c. The affine hull of two distinct points is called a line.

d. A flat is a subspace.

e. A plane in \({\mathbb{R}^3}\) is a hyper plane.

The conditions for affine dependence are stronger than those for linear dependence, so an affinely dependent set is automatically linearly dependent. Also, a linearly independent set cannot be affinely dependent and therefore must be affinely independent. Construct two linearly dependent indexed sets\({S_{\bf{1}}}\)and\({S_{\bf{2}}}\)in\({\mathbb{R}^2}\)such that\({S_{\bf{1}}}\)is affinely dependent and\({S_{\bf{2}}}\)is affinely independent. In each case, the set should contain either one, two, or three nonzero points.

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

10.a. If \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely dependent set in \({\mathbb{R}^n}\), then the set \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) of homogeneous forms may be linearly independent.

b. If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) and \({{\mathop{\rm v}\nolimits} _4}\) are in \({\mathbb{R}^3}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _2} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _4} - {{\mathop{\rm v}\nolimits} _1}} \right\}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) is affinely independent.

c. Given \(S = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}} \right\}\) in \({\mathbb{R}^n}\), each \({\bf{p}}\) in\({\mathop{\rm aff}\nolimits} S\) has a unique representation as an affine combination of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _k}\).

d. When color information is specified at each vertex \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\) of a triangle in \({\mathbb{R}^3}\), then the color may be interpolated at a point p in \({\mathop{\rm aff}\nolimits} \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) using the barycentric coordinates of p.

e. If T is a triangle in \({\mathbb{R}^2}\) and if a point p is on edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.

Question: 1. Let Lbe the line in \({\mathbb{R}^{\bf{2}}}\) through the points \(\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{4}}\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{1}}\end{array}} \right)\). Find a linear functional f and a real number d such that \(L = \left( {f:d} \right)\).

Question 2: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}0\\{ - 1}\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\1\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right)\) in \({\mathbb{R}^{\bf{2}}}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 2{x_1} + {x_2}\)
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