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Chapter 8: Q8.1-6E (page 437)

Question: In Exercises 5 and 6, let \({{\bf{b}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{b}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{ - {\bf{2}}}\end{array}} \right)\), and \({{\bf{b}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{5}}}\\{\bf{1}}\end{array}} \right)\) 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 is 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. \({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{ - {\bf{19}}}\\{\bf{5}}\end{array}} \right)\) b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{{\bf{1}}.{\bf{5}}}\\{ - {\bf{1}}.{\bf{3}}}\\{ - .{\bf{5}}}\end{array}} \right)\) c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{4}}}\\{\bf{0}}\end{array}} \right)\)

Short Answer

Expert verified

a. \({{\bf{p}}_1} = - 4\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 3\left( {{{\bf{b}}_3}} \right) \in {\rm{aff}}\,\,S\), the sum of coefficients of in S is 1.

b. \({{\bf{p}}_2} = 0.2\left( {{{\bf{b}}_1}} \right) + 0.5\left( {{{\bf{b}}_2}} \right) + 0.3\left( {{{\bf{b}}_3}} \right) \in {\rm{aff}}\,S\), the sum of coefficients of in S is 1.

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

Step by step solution

01

Find the augmented matrix

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

\(\begin{array}{c}M = \left( {\begin{array}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_2}}&{{{\bf{b}}_3}}&{{{\bf{p}}_1}}&{{{\bf{p}}_2}}&{{{\bf{p}}_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}2&1&2&0&{1.5}&5\\1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\1&{ - 2}&1&{ - 5}&{ - 0.5}&0\end{array}} \right)\end{array}\)

02

Write the row reduced form of the augmented matrix

Row reduce the augmented matrix as shown below:

\(\begin{array}{c}M = \left( {\begin{array}{*{20}{c}}0&1&{12}&{38}&{4.1}&{13}\\1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&{ - 2}&6&{14}&{0.8}&4\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_1} \to {R_1} - 2{R_2}\\{R_3} \to {R_3} - {R_2}\end{array} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&1&{12}&{38}&{4.1}&{13}\\0&{ - 2}&6&{14}&{0.8}&4\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_1} \leftrightarrow {R_2}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&1&{12}&{38}&{4.1}&{13}\\0&0&{30}&{90}&{9.0}&{30}\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to {R_3} + 2{R_2}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&1&{12}&{38}&{4.1}&{13}\\0&0&1&3&{0.3}&1\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to \frac{1}{{30}}{R_3}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&0&{ - 4}&{0.2}&1\\0&1&0&2&{0.5}&1\\0&0&1&3&{0.3}&1\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_1} \to {R_1} + 5{R_3}\\{R_2} \to {R_2} - 12{R_3}\end{array} \right\}\end{array}\)

03

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

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

\({{\bf{p}}_1} = - 4\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 3\left( {{{\bf{b}}_3}} \right)\)

The sum of the coefficients is \( - 4 + 2 + 3 = 1\).

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

\({{\bf{p}}_1} = - 4\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 3\left( {{{\bf{b}}_3}} \right)\)

04

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

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

\({{\bf{p}}_2} = 0.2\left( {{{\bf{b}}_1}} \right) + 0.5\left( {{{\bf{b}}_2}} \right) + 0.3\left( {{{\bf{b}}_3}} \right)\)

The sum of coefficients is \(0.2 + 0.5 + 0.3 = 1\).

So, \({{\bf{p}}_2}\) is an affine combination of point in S.

\({{\bf{p}}_2} = 0.2\left( {{{\bf{b}}_1}} \right) + 0.5\left( {{{\bf{b}}_2}} \right) + 0.3\left( {{{\bf{b}}_3}} \right)\)

05

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

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

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

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

So, \({{\bf{p}}_3}\) is not an affine combination of point in S.

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Most popular questions from this chapter

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?)

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 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.

Let\(\left\{ {{p_1},{p_2},{p_3}} \right\}\)be an affinely dependent set of points in\({\mathbb{R}^{\bf{n}}}\)and let\(f:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\)be a linear transformation. Show that\(\left\{ {f\left( {{{\bf{p}}_1}} \right),f\left( {{{\bf{p}}_2}} \right),f\left( {{{\bf{p}}_3}} \right)} \right\}\)is affinely dependent in\({\mathbb{R}^{\bf{m}}}\).

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.]
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