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Chapter 8: Q22E (page 437)

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

22. If \(A \subset B\), then \(affA \subset aff B\).

Short Answer

Expert verified

It is proved that \({\rm{aff}}A \subset {\rm{aff}}B\).

Step by step solution

01

Set S is affine

RecallTheorem 2,whichstates that a set \(S\) is affineif and only if every affine combination of points of \(S\) lies in \(S\).

That is, \(S\) is affine if and only if \(S = {\mathop{\rm aff}\nolimits} S\).

02

Show that \(affA \subset affB\)

Suppose that \(B\) contains all affine combinations of points of \(B\), that is \(B \subset {\mathop{\rm aff}\nolimits} B\).And,\(A\) contains every affine combination of points of \(A\), so \(A \subset {\mathop{\rm aff}\nolimits} A\). Thus,\(A \subset B \subset {\mathop{\rm aff}\nolimits} B\).

This implies that,\({\mathop{\rm aff}\nolimits} A \subset {\mathop{\rm aff}\nolimits} B\).

Thus, it is proved that \({\mathop{\rm aff}\nolimits} A \subset {\mathop{\rm aff}\nolimits} B\).

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Most popular questions from this chapter

Suppose \({\bf{y}}\) is orthogonal to \({\bf{u}}\) and \({\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to every \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\). (Hint: An arbitrary \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\) has the form \({\bf{w}} = {c_1}{\bf{u}} + {c_2}{\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to such a vector \({\bf{w}}\).)

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: 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 \({\mathbb{R}^n}\).

Let \({\bf{x}}\left( t \right)\) be a cubic Bézier curve determined by points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\), \({{\bf{p}}_2}\), and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\). Determine how \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have \({\bf{x}}'\left( 1 \right) = 0\)?

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_1}\) as the origin of the coordinate system.]

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.
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