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Chapter 8: Q8.1-10E (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 \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{4}}\end{array}} \right)\). 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 \({{\bf{v}}_1} = \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\) and \({{\bf{v}}_2} = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\2\end{array}} \right)\).

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:

\({\bf{x}} = {x_3}\left( {\begin{array}{*{20}{c}}5\\1\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\)

02

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

Substitute 0 for \({x_3}\) in the equation \({\bf{x}} = {x_3}\left( {\begin{array}{*{20}{c}}5\\1\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\) to find \({{\bf{v}}_1}\) as shown below:

\(\begin{array}{c}{{\bf{v}}_1} = 0\left( {\begin{array}{*{20}{c}}5\\1\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\end{array}\)

Substitute 1 for \({x_3}\) in the equation \({\bf{x}} = {x_3}\left( {\begin{array}{*{20}{c}}5\\1\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\) to find \({{\bf{v}}_2}\) as shown below:

\(\begin{array}{c}{{\bf{v}}_1} = 1\left( {\begin{array}{*{20}{c}}5\\1\\{ - 2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\2\end{array}} \right)\end{array}\)

So, one of the points are \({{\bf{v}}_1} = \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\4\end{array}} \right)\) and \({{\bf{v}}_2} = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\2\end{array}} \right)\).

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Most popular questions from this chapter

Let \(W = \left\{ {{{\bf{v}}_1},......,{{\bf{v}}_p}} \right\}\). Show that if \({\bf{x}}\) is orthogonal to each \({{\bf{v}}_j}\), for \(1 \le j \le p\), then \({\bf{x}}\) is orthogonal to every vector in \(W\).

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: 16. Let \({\rm{v}} \in {\mathbb{R}^n}\)and let \(k \in \mathbb{R}\). Prove that \(S = \left\{ {{\rm{x}} \in {\mathbb{R}^n}:{\rm{x}} \cdot {\rm{v}} = k} \right\}\)is an affine subset of \({\mathbb{R}^n}\).

Question 4: Repeat Exercise 2 where \(m\) is the minimum value of \(f\) on \(S\) instead of the maximum value.

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 \({\mathbb{R}^m}\).
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