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Chapter 8: Q5E (page 437)

In Exercises 5 and 6, let \({{\bf{b}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right)\), \({{\bf{b}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{ - {\bf{2}}}\end{aligned}} \right)\), and \({{\bf{b}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{ - {\bf{5}}}\\{\bf{1}}\end{aligned}} \right)\) and \(S = \left\{ {{{\bf{b}}_{\bf{1}}},\,{{\bf{b}}_{\bf{2}}},\,{{\bf{b}}_{\bf{3}}}} \right\}\). Note that S is an orthogonal basis of \({\mathbb{R}^{\bf{3}}}\). Write each of the given points as an affine combination of the points in the set S, if possible. (Hint: Use Theorem 5 in section 6.2 instead of row reduction to find the weights.)

a. \({{\bf{p}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{8}}\\{\bf{4}}\end{aligned}} \right)\)

b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{6}}\\{ - {\bf{3}}}\\{\bf{3}}\end{aligned}} \right)\)

c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{ - {\bf{1}}}\\{ - {\bf{5}}}\end{aligned}} \right)\)

Short Answer

Expert verified

a. \({{\bf{p}}_1} = 3\left( {{{\bf{b}}_1}} \right) - 1\left( {{{\bf{b}}_2}} \right) - 1\left( {{{\bf{b}}_3}} \right) \in {\rm{aff}}\,\,S\), the sum of coefficients of in S 1.

b. \({{\bf{p}}_2} = 2\left( {{{\bf{b}}_1}} \right) + 0\left( {{{\bf{b}}_2}} \right) + 1\left( {{{\bf{b}}_3}} \right) \notin {\rm{aff}}\,S\), the sum of coefficients of in S is not 1.

c. \({{\bf{p}}_3} = - 1\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 0\left( {{{\bf{b}}_3}} \right) \in {\rm{aff}}\,S\), the sum of coefficients of in S 1.

Step by step solution

01

Find the augmented matrix

Write the augmented matrix by using the given points as shown below:

\(\begin{aligned}{c}M = \left( {\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_2}}&{{{\bf{b}}_3}}&{{{\bf{p}}_1}}&{{{\bf{p}}_2}}&{{{\bf{p}}_3}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}2&1&2&3&6&0\\1&0&{ - 5}&8&{ - 3}&{ - 1}\\1&{ - 2}&1&4&3&{ - 5}\end{aligned}} \right)\end{aligned}\)

02

Obtain the row reduced form of the augmented matrix

Write the augmented matrix as shown below:

\(\begin{aligned}{c}M = \left( {\begin{aligned}{*{20}{c}}0&1&{12}&{ - 13}&{12}&2\\1&0&{ - 5}&8&{ - 3}&{ - 1}\\0&{ - 2}&6&{ - 4}&6&{ - 4}\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{aligned}{l}{R_1} \to {R_1} - 2{R_2}\\{R_3} \to {R_3} - {R_2}\end{aligned} \right\}\\ = \left( {\begin{aligned}{*{20}{c}}1&0&{ - 5}&8&{ - 3}&{ - 1}\\0&1&{12}&{ - 13}&{12}&2\\0&{ - 2}&6&4&6&{ - 4}\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_1} \leftrightarrow {R_2}} \right\}\\ = \left( {\begin{aligned}{*{20}{c}}1&0&{ - 5}&8&{ - 3}&{ - 1}\\0&1&{12}&{ - 13}&{12}&2\\0&1&{ - 3}&2&{ - 3}&2\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to - \frac{1}{2}{R_3}} \right\}\\ = \left( {\begin{aligned}{*{20}{c}}1&0&{ - 5}&8&{ - 3}&{ - 1}\\0&1&{12}&{ - 13}&{12}&2\\0&0&{ - 15}&{15}&{ - 15}&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to {R_3} - {R_2}} \right\}\\ = \left( {\begin{aligned}{*{20}{c}}1&0&{ - 5}&8&{ - 3}&{ - 1}\\0&1&{12}&{ - 13}&{12}&2\\0&0&1&{ - 1}&1&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to - \frac{1}{{15}}{R_3}} \right\}\\ = \left( {\begin{aligned}{*{20}{c}}1&0&0&3&2&{ - 1}\\0&1&0&{ - 1}&0&2\\0&0&1&{ - 1}&1&0\end{aligned}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{aligned}{l}{R_1} \to {R_1} + 5{R_3}\\{R_2} \to {R_2} - 12{R_3}\end{aligned} \right\}\end{aligned}\)

03

Check for the affine combination of \({{\bf{p}}_{\bf{1}}}\)

Use the augmented matrix \({{\bf{p}}_1}\) that can be expressed as shown below:

\({{\bf{p}}_1} = 3\left( {{{\bf{b}}_1}} \right) - 1\left( {{{\bf{b}}_2}} \right) - 1\left( {{{\bf{b}}_3}} \right)\)

The sum of coefficients is \(3 - 1 - 1 = 1\).

Thus, \({{\bf{p}}_1}\) is an affine combination of point in set S.

\({{\bf{p}}_1} = 3\left( {{{\bf{b}}_1}} \right) - 1\left( {{{\bf{b}}_2}} \right) - 1\left( {{{\bf{b}}_3}} \right)\)

04

Check for the affine combination of \({{\bf{p}}_{\bf{2}}}\)

Use the augmented matrix \({{\bf{p}}_2}\) that can be expressed as shown below:

\({{\bf{p}}_2} = 2\left( {{{\bf{b}}_1}} \right) + 0\left( {{{\bf{b}}_2}} \right) + 1\left( {{{\bf{b}}_3}} \right)\)

The sum of the coefficients is \(2 + 0 + 1 = 3 \ne 1\).

So, \({{\bf{p}}_2}\) is not an affine combination of point in set S.

05

Check for the affine combination of \({{\bf{p}}_{\bf{3}}}\)

Use the augmented matrix \({{\bf{p}}_3}\) that can be expressed as shown below:

\({{\bf{p}}_3} = - 1\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 0\left( {{{\bf{b}}_3}} \right)\)

The sum of the coefficients is \( - 1 + 2 + 0 = 1\).

So, \({{\bf{p}}_3}\) is an affine combination of point in set S.

\({{\bf{p}}_3} = - 1\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 0\left( {{{\bf{b}}_3}} \right)\)

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Most popular questions from this chapter

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{5}}\end{array}} \right]\),\({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{3}}\end{array}} \right]\),\({p_1} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{5}}\end{array}} \right]\),\({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\end{array}} \right]\),\({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{3}}\end{array}} \right]\),\({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{0}}\end{array}} \right]\),\({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\),\({p_{\bf{6}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\),\({p_{\bf{7}}} = \left[ {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{4}}\end{array}} \right]\)and let\(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of\({p_1}\),\({p_{\bf{2}}}\), and\({p_{\bf{3}}}\)with respect to S.
  3. On graph paper, sketch the triangle\(T\)with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\), extend the sides as in Figure 8, and plot the points\({p_{\bf{4}}}\),\({p_{\bf{5}}}\),\({p_{\bf{6}}}\), and\({p_{\bf{7}}}\). Without calculating the actual values, determine the signs of the barycentric coordinates of points\({p_{\bf{4}}}\),\({p_{\bf{5}}}\),\({p_{\bf{6}}}\), and\({p_{\bf{7}}}\).

Question: 13. Suppose \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\). Show that Span \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) is a plane in \({\mathbb{R}^3}\). (Hint: What can you say about \({\rm{u}}\) and \({\rm{v}}\)when Span \(\left\{ {{\rm{u,v}}} \right\}\) is a plane?)

Let \({\bf{x}}\left( t \right)\) be a B-spline in Exercise 2, with control points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\) , \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\) and determine how the derivatives \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points. Give geometric descriptions of the directions of these tangent vectors. Explore what happens when both \({\bf{x}}'\left( 0 \right)\)and \({\bf{x}}'\left( 1 \right)\)equal 0. Justify your assertions.

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 \({{\bf{p}}_2}\) as the origin of the coordinate system.]

Question 2: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}0\\{ - 1}\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\1\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right)\) in \({\mathbb{R}^{\bf{2}}}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 2{x_1} + {x_2}\)

In Exercises 11 and 12, mark each statement True or False. Justify each answer.

12.a. The essential properties of Bezier curves are preserved under the action of linear transformations, but not translations.

b. When two Bezier curves \({\mathop{\rm x}\nolimits} \left( t \right)\) and \(y\left( t \right)\) are joined at the point where \({\mathop{\rm x}\nolimits} \left( 1 \right) = y\left( 0 \right)\), the combined curve has \({G^0}\) continuity at that point.

c. The Bezier basis matrix is a matrix whose columns are the control points of the curve.
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