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Chapter 8: Q7E (page 437)

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

Short Answer

Expert verified
  1. The normal vector is \(n = \left( {\begin{array}{*{20}{c}}0\\2\\3\end{array}} \right)\) or a multiple
  2. The linear function is \(f\left( x \right) = 2{x_2} + 3{x_3}\) , and the real number is \(d = 11\).

Step by step solution

01

Write the given data

Let \({v_1} = \left( {\begin{array}{*{20}{c}}1\\1\\3\end{array}} \right)\), \({v_2} = \left( {\begin{array}{*{20}{c}}2\\4\\1\end{array}} \right)\), and \({v_3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 2}\\5\end{array}} \right)\).

Then, \({v_2} - {v_1} = \left( {\begin{array}{*{20}{c}}1\\3\\{ - 2}\end{array}} \right),\) and \({v_3} - {v_1} = \left( {\begin{array}{*{20}{c}}{ - 2}\\{ - 3}\\2\end{array}} \right)\).

02

Use the cross product to compute n

(a)

\(\begin{array}{c}n = \left( {{v_2} - {v_1}} \right) \times \left( {{v_3} - {v_1}} \right)\\ = \left| {\begin{array}{*{20}{c}}1&{ - 2}&i\\3&{ - 3}&j\\{ - 2}&2&k\end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}}3&{ - 3}\\{ - 2}&2\end{array}} \right|i - \left| {\begin{array}{*{20}{c}}1&{ - 2}\\{ - 2}&2\end{array}} \right|j + \left| {\begin{array}{*{20}{c}}1&{ - 2}\\3&{ - 3}\end{array}} \right|k\\ = 0i + 2j + 3k\end{array}\)

Thus, the normal vector is \(n = \left( {\begin{array}{*{20}{c}}0\\2\\3\end{array}} \right)\).

03

Find a linear functional f and a real number d

(b)

Using part (a) to obtain the linear functional f as shown below:

\(\begin{array}{c}f\left( x \right) = n \cdot x\\ = \left( {\begin{array}{*{20}{c}}0&2&3\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\f\left( x \right) = 2{x_2} + 3{x_3}\end{array}\)

Note that, \({v_i}\) in \(H = \left( {f:d} \right)\) such that, \(f\left( {{v_i}} \right) = d\) for \(i = 1,2,3\).

\(\begin{array}{c}d = f\left( {{v_1}} \right)\\ = f\left( {1,1,3} \right)\\ = 2\left( 1 \right) + 3\left( 3 \right)\\ = 2 + 9\\d = 11\end{array}\)

Thus, the linear function is \(f\left( x \right) = 2{x_2} + 3{x_3}\) , and the real number is \(d = 11\).

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Most popular questions from this chapter

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 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

3.\(\left( {\begin{aligned}{{}}1\\2\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 2}\\{ - 4}\\8\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\{ - 1}\\{11}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\{15}\\{ - 9}\end{aligned}} \right)\)

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{2}}}\\{\bf{2}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{4}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{aligned}} \right)\)

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

1.\(\left( {\begin{aligned}{{}}3\\{ - 3}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\6\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\0\end{aligned}} \right)\)

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 \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}\\{ - {\bf{4}}}&{\bf{2}}\\{\bf{7}}&{ - {\bf{6}}}\end{array}} \right)\). That is, \(H = {\bf{Col}}\,B\).(Hint: How is \({\bf{Col}}\,B\) related to Nul \({B^T}\)? See section 6.1)
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