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Chapter 8: Q8E (page 437)

Question: In Exercise 8, 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 \(H = \left( {f:d} \right)\).

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

Short Answer

Expert verified
  1. The normal vector is \(n = \left( {\begin{array}{*{20}{c}}4\\3\\{ - 6}\end{array}} \right)\) or a multiple
  2. The linear functional is \(f\left( x \right) = 4{x_1} + 3{x_2} - 6{x_3}\) , and the real number is \(d = - 8\).

Step by step solution

01

Write the given data

Let \({v_1} = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right)\), \({v_2} = \left( {\begin{array}{*{20}{c}}4\\{ - 2}\\3\end{array}} \right)\), and \({v_3} = \left( {\begin{array}{*{20}{c}}7\\{ - 4}\\4\end{array}} \right)\).

Then, the vectors are \({v_2} - {v_1} = \left( {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right),\) and \({v_3} - {v_1} = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\3\end{array}} \right)\).

02

Use the cross product to compute n

(a)

\(\begin{array}{c}n = \left( {{v_2} - {v_1}} \right) \times \left( {{v_3} - {v_1}} \right)\\ = \left| {\begin{array}{*{20}{c}}3&6&i\\0&{ - 2}&j\\2&3&k\end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}}0&{ - 2}\\2&3\end{array}} \right|i - \left| {\begin{array}{*{20}{c}}3&6\\2&3\end{array}} \right|j + \left| {\begin{array}{*{20}{c}}3&6\\0&{ - 2}\end{array}} \right|k\\ = 4i + 3j - 6k\end{array}\)

Thus, the normal vector is \(n = \left( {\begin{array}{*{20}{c}}4\\3\\{ - 6}\end{array}} \right)\).

03

Find a linear functional f and a real number d

(b)

Using part (a), the linear functional f can be obtained as shown below:

\(\begin{array}{c}f\left( x \right) = n \cdot x\\ = \left[ {\begin{array}{*{20}{c}}4&3&{ - 6}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\f\left( x \right) = 4{x_1} + 3{x_2} - 6{x_3}\end{array}\)

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

\(\begin{array}{c}d = f\left( {{v_1}} \right)\\ = f\left( {1, - 2,1} \right)\\ = 4\left( 1 \right) + 3\left( { - 2} \right) - 6\left( 1 \right)\\ = 4 - 6 - 6\\d = - 8\end{array}\)

Thus, the linear function is \(f\left( x \right) = 4{x_1} + 3{x_2} - 6{x_3}\) , and the real number is \(d = - 8\).

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Most popular questions from this chapter

Question: 18. Choose a set \(S\) of four points in \({\mathbb{R}^3}\) such that aff \(S\) is the plane \(2{x_1} + {x_2} - 3{x_3} = 12\). Justify your work.

Question: 27. Give an example of a closed subset\(S\)of\({\mathbb{R}^{\bf{2}}}\)such that\({\rm{conv}}\,S\)is not closed.

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.

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. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{4}}}\\{\bf{0}}\end{array}} \right)\)

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