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Chapter 8: Q-8.2-20E (page 437)

Suppose that\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)is an affinely independent set in\({\mathbb{R}^{\bf{n}}}\)and q is an arbitrary point in\({\mathbb{R}^{\bf{n}}}\). Show that the translated set\(\left\{ {{p_1} + q,{p_2} + q,{p_3} + {\bf{q}}} \right\}\)is also affinely independent.

Short Answer

Expert verified

The translated set is also affinely independent.

Step by step solution

01

State the condition for affinely dependence

The set is said to be affinely dependent if the set \(\left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}},...,{{\bf{v}}_p}} \right\}\) in the dimension \({\mathbb{R}^n}\) exists such that for non-zero scalars \({c_1},{c_2},...,{c_p}\), the sum of scalars is zero i.e. \({c_1} + {c_2} + ... + {c_p} = 0\), and \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\).

02

Show affinely dependence 

The set is affinely dependent if the set\(\left\{ {{p_1} + q,{p_2} + q,{p_3} + {\bf{q}}} \right\}\)in\({\mathbb{R}^n}\)exists such that for non-zero scalars\({c_1},{c_2}\)and\({c_3}\)the is zero i.e.\({c_1} + {c_2} + {c_3} = 0\).

So,\({c_1}\left( {{p_1} + q} \right) + {c_2}\left( {{p_2} + q} \right) + {c_3}\left( {{p_3} + q} \right) = 0\).

Simplify\({c_1}\left( {{p_1} + q} \right) + {c_2}\left( {{p_2} + q} \right) + {c_3}\left( {{p_3} + q} \right) = 0\)as shown below:

\(\begin{array}{c}{c_1}{p_1} + {c_1}q + {c_2}{p_2} + {c_2}q + {c_3}{p_3} + {c_3}q = 0\\{c_1}{p_1} + {c_2}{p_2} + {c_3}{p_3} + \left( {{c_1} + {c_2} + {c_3}} \right){\bf{q}} = 0\end{array}\)

As\({c_1} + {c_2} + {c_3} = 0\),\({c_1}{p_1} + {c_2}{p_2} + {c_3}{p_3} = 0\).

Thus,\({c_1}{{\bf{p}}_1} + {c_2}{{\bf{p}}_2} + {c_3}{{\bf{p}}_3} = 0\)and\({c_1} + {c_2} + {c_3} = 0\)show that\(S = \left\{ {{p_1},{p_2},{p_3}} \right\}\) is an affinely dependent set of points in \({\mathbb{R}^n}\).

But\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)is affinely independent. Thus, the translated set is also affinely independent.

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Most popular questions from this chapter

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?

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 \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{3}}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{4}}\\{ - {\bf{2}}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{6}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{{\bf{17}}}\\{\bf{1}}\\{\bf{5}}\end{aligned}} \right)\)

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let A be the \({\bf{1}} \times {\bf{4}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{2}}}\end{array}} \right)\) and let \(b = {\bf{5}}\). Let \(H = \left\{ {{\bf{x}}\,\,{\rm{in}}\,{\mathbb{R}^{\bf{4}}}:A{\bf{x}} = {\bf{b}}} \right\}\).

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.
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