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Chapter 8: Q8.1-9E (page 437)

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 \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{0}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).

Short Answer

Expert verified

The points are \({{\bf{v}}_1} = \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\) and \({{\bf{v}}_2} = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right)\).

Step by step solution

01

Substitute the given values in the equation of x

Substitute the given values in the equation \({\bf{x}} = {x_3}{\bf{u}} + {\bf{p}}\) as shown below:

\({\bf{x}} = {x_3}\left( {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\)

02

Find the value of \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\)

Substitute 0 for \({x_3}\) in the equation \({\bf{x}} = {x_3}\left( {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\) to find \({{\bf{v}}_1}\) as shown below:

\(\begin{array}{c}{{\bf{v}}_1} = 0\left( {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\end{array}\)

Substitute 1 for \({x_3}\) in the equation \({\bf{x}} = {x_3}\left( {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\) to find \({{\bf{v}}_2}\) as shown below:

\(\begin{array}{c}{{\bf{v}}_2} = 1\left( {\begin{array}{*{20}{c}}4\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right)\end{array}\)

So, \({{\bf{v}}_1} = \left( {\begin{array}{*{20}{c}}{ - 3}\\0\end{array}} \right)\) and \({{\bf{v}}_2} = \left( {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right)\).

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Most popular questions from this chapter

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

22. If \(A \subset B\), then \(affA \subset aff B\).

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

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

Question: 19. Let \(S\) be an affine subset of \({\mathbb{R}^n}\) , suppose \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation, and let \(f\left( S \right)\) denote the set of images \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). Prove that \(f\left( S \right)\)is an affine subset of \({\mathbb{R}^m}\).

Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).
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