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Chapter 8: Q18E (page 437)

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 plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{4}}&{ - {\bf{5}}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{8}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\).

Short Answer

Expert verified

The linear functional is \(f\left( {{x_1},{x_2},{x_3}} \right) = - 11{x_1} + 4{x_2} + {x_3}\) and \(d = 0\).

Step by step solution

01

Write the matrix equation

The matrix equation can be written as follows:

\(\begin{array}{c}B{\bf{x}} = 0\\\left( {\begin{array}{*{20}{c}}1&4&{ - 5}\\0&{ - 2}&8\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right)\end{array}\)

02

Write the equation using matrix multiplication

The matrix equation can be simplified as shown below:

\({x_1} + 4{x_2} - 5{x_3} = 0\)

And,

\(\begin{array}{c} - 2{x_2} + 8{x_3} = 0\\{x_2} = 4{x_3}\end{array}\)

Simplifying the above equations:

\(\begin{array}{c}{x_1} + 4\left( {4{x_3}} \right) - 5{x_3} = 0\\{x_1} = - 11{x_3}\end{array}\)

03

Write the general solution

The general solution is:

\(\begin{array}{c}{\bf{x}} = \left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{ - 11{x_3}}\\{4{x_3}}\\{{x_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{ - 11}\\4\\1\end{array}} \right){x_3}\end{array}\)

So, \(f\left( {{x_1},{x_2},{x_3}} \right) = - 11{x_1} + 4{x_2} + {x_3}\) and \(d = 0\).

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Most popular questions from this chapter

Question: In Exercises 5-8, find the minimal representation of the polytope defined by the inequalities \(A{\bf{x}} \le {\bf{b}}\) and \({\bf{x}} \ge {\bf{0}}\).

5. \(A = \left( {\begin{array}{*{20}{c}}1&2\\3&1\end{array}} \right),{\rm{ }}{\bf{b}} = \left( {\begin{array}{*{20}{c}}{{\bf{10}}}\\{{\bf{15}}}\end{array}} \right)\)

In Exercises 21鈥�24, a, b, and c are noncollinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \)and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

24. Take q on the line segment from b to c and consider the line through q and a, which may be written as\(p = \left( {1 - x} \right)q + xa\)for all real x. Show that, for each x,\(det\left[ {\begin{array}{*{20}{c}}{\widetilde p}&{\widetilde b}&{\widetilde c}\end{array}} \right] = x \cdot det\left[ {\begin{array}{*{20}{c}}{\widetilde a}&{\widetilde b}&{\widetilde c}\end{array}} \right]\). From this and earlier work, conclude that the parameter x is the first barycentric coordinate of p. However, by construction, the parameter x also determines the relative distance between p and q along the segment from q to a. (When x = 1, p = a.) When this fact is applied to Example 5, it shows that the colors at vertex a and the point q are smoothly interpolated as p moves along the line between a and q.

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.

Question: Let \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{3}}\\{\bf{0}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{1}}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\)

\({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{\bf{5}}\\{\bf{3}}\end{array}} \right)\) b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{9}}}\\{{\bf{10}}}\\{\bf{9}}\\{ - {\bf{13}}}\end{array}} \right)\) c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{2}}\\{\bf{8}}\\{\bf{5}}\end{array}} \right)\)

and \(S = \left\{ {{{\bf{v}}_1},\,\,{{\bf{v}}_2},\,{{\bf{v}}_3}} \right\}\). It can be shown that S is linearly independent.

a. Is \({{\bf{p}}_{\bf{1}}}\) is span S? Is \({{\bf{p}}_{\bf{1}}}\) is \({\bf{aff}}\,S\)?

b. Is \({{\bf{p}}_{\bf{2}}}\) is span S? Is \({{\bf{p}}_{\bf{2}}}\) is \({\bf{aff}}\,S\)?

c. Is \({{\bf{p}}_{\bf{3}}}\) is span S? Is \({{\bf{p}}_{\bf{3}}}\) is \({\bf{aff}}\,S\)?

Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{4}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).
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