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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

In Exercises 13-15 concern the subdivision of a Bezier curve shown in Figure 7. Let \({\mathop{\rm x}\nolimits} \left( t \right)\) be the Bezier curve, with control points \({{\mathop{\rm p}\nolimits} _0},...,{{\mathop{\rm p}\nolimits} _3}\), and let \({\mathop{\rm y}\nolimits} \left( t \right)\) and \({\mathop{\rm z}\nolimits} \left( t \right)\) be the subdividing Bezier curves as in the text, with control points \({{\mathop{\rm q}\nolimits} _0},...,{{\mathop{\rm q}\nolimits} _3}\) and \({{\mathop{\rm r}\nolimits} _0},...,{{\mathop{\rm r}\nolimits} _3}\), respectively.

15. Sometimes only one-half of a Bezier curve needs further subdividing. For example, subdivision of the “left” side is accomplished with parts (a) and (c) of Exercise 13 and equation (8). When both halves of the curve \({\mathop{\rm x}\nolimits} \left( t \right)\) are divided, it is possible to organize calculations efficiently to calculate both left and right control points concurrently, without using equation (8) directly.

a. Show that the tangent vector \(y'\left( 1 \right)\) and \(z'\left( 0 \right)\) are equal.

b. Use part (a) to show that \({{\mathop{\rm q}\nolimits} _3}\) (which equals \({{\mathop{\rm r}\nolimits} _0}\)) is the midpoint of the segment from \({{\mathop{\rm q}\nolimits} _2}\) to \({{\mathop{\rm r}\nolimits} _1}\).

c. Using part (b) and the results of Exercises 13 and 14, write an algorithm that computes the control points for both \({\mathop{\rm y}\nolimits} \left( t \right)\) and \({\mathop{\rm z}\nolimits} \left( t \right)\) in an efficient manner. The only operations needed are sums and division by 2.

Let \({\bf{x}}\left( t \right)\) and \({\bf{y}}\left( t \right)\) be cubic Bézier curves with control points \(\left\{ {{{\bf{p}}_{\bf{o}}}{\bf{,}}{{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}}} \right\}\)and \(\left\{ {{{\bf{p}}_{\bf{3}}}{\bf{,}}{{\bf{p}}_{\bf{4}}}{\bf{,}}{{\bf{p}}_{\bf{5}}}{\bf{,}}{{\bf{p}}_{\bf{6}}}} \right\}\) respectively, so that \({\bf{x}}\left( t \right)\) and \({\bf{y}}\left( t \right)\) are joined at \({{\bf{p}}_3}\) . The following questions refer to the curve consisting of \({\bf{x}}\left( t \right)\) followed by \(y\left( t \right)\). For simplicity, assume that the curve is in \({\mathbb{R}^2}\).

a. What condition on the control points will guarantee that the curve has \({C^1}\) continuity at \({{\bf{p}}_3}\) ? Justify your answer.

b. What happens when \({\bf{x'}}\left( 1 \right)\) and \({\bf{y'}}\left( 1 \right)\) are both the zero vector?

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)

Let\(T\)be a tetrahedron in “standard” position, with three edges along the three positive coordinate axes in\({\mathbb{R}^3}\), and suppose the vertices are\(a{{\bf{e}}_1}\),\(b{{\bf{e}}_2}\),\(c{{\bf{e}}_{\bf{3}}}\), and 0, where\(\left[ {\begin{array}{*{20}{c}}{{{\bf{e}}_1}}&{{{\bf{e}}_2}}&{{{\bf{e}}_3}}\end{array}} \right] = {I_3}\). Find formulas for the barycentric coordinates of an arbitrary point\({\bf{p}}\)in\({\mathbb{R}^3}\).

Questions: Let \({F_{\bf{1}}}\) and \({F_{\bf{2}}}\) be 4-dimensional flats in \({\mathbb{R}^{\bf{6}}}\), and suppose that \({F_{\bf{1}}} \cap {F_{\bf{2}}} \ne \phi \). What are the possible dimension of \({F_{\bf{1}}} \cap {F_{\bf{2}}}\)?
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