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Chapter 8: Q16E (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 A be the \({\bf{1}} \times {\bf{5}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{2}}&{\bf{5}}&{ - {\bf{3}}}&{\bf{0}}&{\bf{6}}\end{array}} \right)\). Note that \({\bf{Nul}}\,\,A\) is in \({\mathbb{R}^{\bf{5}}}\). Let \(H = {\bf{Nul}}\,\,A\).

Short Answer

Expert verified

\(f\left( {{x_1},{x_2},{x_3},{x_4},{x_5}} \right) = 2{x_1} + 5{x_2} - 3{x_3} + 6{x_5}\) and \(d = 0\)

Step by step solution

01

Write the matrix equation

The matrix equation can be written as follows:

\(\begin{array}{c}A{\bf{x}} = 0\\\left( {\begin{array}{*{20}{c}}2&5&{ - 3}&0&6\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\\{{x_5}}\end{array}} \right) = 0\end{array}\)

02

Write the equation using matrix multiplication

The matrix equation is shown below:

\(\begin{array}{c}2{x_1} + 5{x_2} - 3{x_3} + 0 + 6{x_5} = 0\\2{x_1} + 5{x_2} - 3{x_3} + 6{x_5} = 0\end{array}\)

So, \(f\left( {{x_1},{x_2},{x_3},{x_4},{x_5}} \right) = 2{x_1} + 5{x_2} - 3{x_3} + 6{x_5}\) and \(d = 0\).

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Most popular questions from this chapter

Question: In Exercise 7, 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 \(H = \left( {f:d} \right)\).

7. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{4}}\\{\bf{1}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{ - {\bf{2}}}\\{\bf{5}}\end{array}} \right)\)

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9.

a. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _1} - {{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _2}} \right\}\) is linearly dependent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent. (Read this carefully.)

b. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set of homogeneous forms \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent.

c. A finite set of points \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is affinely dependent if there exist real numbers \({c_1},...,{c_k}\) , not all zero, such that \({c_1} + ... + {c_k} = 1\) and \({c_1}{{\mathop{\rm v}\nolimits} _1} + ... + {c_k}{{\mathop{\rm v}\nolimits} _k} = 0\).

d. If \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely independent set in \({\mathbb{R}^n}\) and if p in \({\mathbb{R}^n}\) has a negative barycentric coordinate determined by S, then p is not in \({\mathop{\rm aff}\nolimits} S\).

e.

If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},a,\) and \(b\) are in \({\mathbb{R}^3}\) and if ray \({\mathop{\rm a}\nolimits} + t{\mathop{\rm b}\nolimits} \) for \(t \ge 0\) intersects the triangle with vertices \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\) then the barycentric coordinates of the intersection points are all nonnegative.

Question: 26. Let \({\rm{q}} = \left( \begin{array}{l}2\\3\end{array} \right)\), \({\rm{p}} = \left( \begin{array}{l}6\\1\end{array} \right)\). Find a hyperplane \(\left( {f:d} \right)\) that strictly separates \(B\left( {{\rm{q}},3} \right)\) and \(B\left( {{\rm{p}},1} \right)\).

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

a. If \(S = \left\{ {\bf{x}} \right\}\), then \({\rm{aff}}\,S\) is the empty set.

b. A set is affine if and only if it contains its affine hull.

c. A flat of dimension 1 is called a line.

d. A flat of dimension 2 is called a hyper plane.

e. A flat through the origin is a subspace.

TrueType fonts, created by Apple Computer and Adobe Systems, use quadratic Bezier curves, while PostScript fonts, created by Microsoft, use cubic Bezier curves. The cubic curves provide more flexibility for typeface design, but it is important to Microsoft that every typeface using quadratic curves can be transformed into one that used cubic curves. Suppose that \({\mathop{\rm w}\nolimits} \left( t \right)\) is a quadratic curve, with control points \({{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},\) and \({{\mathop{\rm p}\nolimits} _2}\).

  1. Find control points \({{\mathop{\rm r}\nolimits} _0},{{\mathop{\rm r}\nolimits} _1},{{\mathop{\rm r}\nolimits} _2},\), and \({{\mathop{\rm r}\nolimits} _3}\) such that the cubic Bezier curve \({\mathop{\rm x}\nolimits} \left( t \right)\) with these control points has the property that \({\mathop{\rm x}\nolimits} \left( t \right)\) and \({\mathop{\rm w}\nolimits} \left( t \right)\) have the same initial and terminal points and the same tangent vectors at \(t = 0\)and\(t = 1\). (See Exercise 16.)
  1. Show that if \({\mathop{\rm x}\nolimits} \left( t \right)\) is constructed as in part (a), then \({\mathop{\rm x}\nolimits} \left( t \right) = {\mathop{\rm w}\nolimits} \left( t \right)\) for \(0 \le t \le 1\).
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