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Chapter 8: Q25E (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).

25. \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

Short Answer

Expert verified

It is proved that \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\).

Step by step solution

01

Set S is affine

Theorem 2states 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 \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

Let \(x \in {\mathop{\rm aff}\nolimits} \left( {A \cap B} \right)\,\).

Then, \({\mathop{\rm x}\nolimits} \) can be written as the affine combination of the intersection of \(A\), and\(B\). So, \(x = {c_1}{f_1} + ... + {c_k}{f_k}\) are the elements of the intersection of \(A \cap B\), where \({c_1} + ... + {c_k} = 1\).

This means that there exist \({\mathop{\rm aff}\nolimits} A\) and \({\mathop{\rm aff}\nolimits} B\), \({f_i} \in A\,\,{\mathop{\rm and}\nolimits} \,\,\,B\,\,\,{\mathop{\rm for}\nolimits} \,\,{\mathop{\rm all}\nolimits} \,\,i\).

This implies that\({f_i} \in {\mathop{\rm aff}\nolimits} A\,\,\,{\mathop{\rm and}\nolimits} \,\,{\mathop{\rm aff}\nolimits} \,B\) for all \(i\). Therefore, \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\).

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

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Most popular questions from this chapter

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: In Exercise 8, let Hbe the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{2}}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{7}}\\{ - {\bf{4}}}\\{\bf{4}}\end{array}} \right)\)

Find an example in \({\mathbb{R}^2}\) to show that equality need not hold in the statement of Exercise 23.

Question: In Exercise 5, determine whether or not each set is compact and whether or not it is convex.

5. Use the sets from Exercise 3.

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

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