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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

Question 3: Repeat Exercise 1 where \(m\) is the minimum value of f on \(S\) instead of the maximum value.

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

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 \({\mathbb{R}^m}\).

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 H be the column space of the matrix \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}\\{\bf{5}}&{\bf{2}}\\{ - {\bf{4}}}&{ - {\bf{4}}}\end{array}} \right)\). That is, \(H = {\bf{Col}}\,B\).(Hint: How is \({\bf{Col}}\,B\)related to Nul \({B^T}\)? See section 6.1)

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?
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