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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 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. \(f\left( {{x_1},{x_2}} \right) = - 2{x_1} + {x_2}\)

Question: In Exercise 4, determine whether each set is open or closed or neither open nor closed.

4. a. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} = {\bf{1}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} > {\bf{1}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} \le {\bf{1}}\,\,\,and\,\,y > {\bf{0}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):y \ge {x^{\bf{2}}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):y < {x^{\bf{2}}}} \right\}\)

Question: 24. Repeat Exercise 23 for \({v_1} = \left( \begin{array}{l}1\\2\end{array} \right)\), \({v_2} = \left( \begin{array}{l}5\\1\end{array} \right)\), \({v_3} = \left( \begin{array}{l}4\\4\end{array} \right)\) and \(p = \left( \begin{array}{l}2\\3\end{array} \right)\).

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

6. \(A = \left( {\begin{array}{*{20}{c}}2&3\\4&1\end{array}} \right),{\rm{ }}{\mathop{\rm b}\nolimits} = \left( {\begin{array}{*{20}{c}}{18}\\{16}\end{array}} \right)\)

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 A be the \({\bf{1}} \times {\bf{4}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{2}}}\end{array}} \right)\) and let \(b = {\bf{5}}\). Let \(H = \left\{ {{\bf{x}}\,\,{\rm{in}}\,{\mathbb{R}^{\bf{4}}}:A{\bf{x}} = {\bf{b}}} \right\}\).
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