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Chapter 4: Q19E (page 191)

[M] Let

\(P = \left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{2}}&{ - {\bf{1}}}\\{ - {\bf{3}}}&{ - {\bf{5}}}&{\bf{0}}\\{\bf{4}}&{\bf{6}}&{\bf{1}}\end{array}} \right]\), \({{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{2}}\\{\bf{3}}\end{array}} \right],\,{{\bf{v}}_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{8}}}\\{\bf{5}}\\{\bf{2}}\end{array}} \right],\,{{\bf{v}}_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{7}}}\\{\bf{2}}\\{\bf{6}}\end{array}} \right]\)

a. Find a basis \(\left\{ {{{\bf{u}}_{\bf{1}}},\,{{\bf{u}}_{\bf{2}}},\,{{\bf{u}}_{\bf{3}}}} \right\}\) for \({\mathbb{R}^{\bf{3}}}\) such that P is the change of coordinates matrix from \(\left\{ {{{\bf{u}}_{\bf{1}}},\,{{\bf{u}}_{\bf{2}}},\,{{\bf{u}}_{\bf{3}}}} \right\}\) to the basis \(\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}},\,{{\bf{v}}_{\bf{3}}}} \right\}\).[Hint: What do the columns of \(\mathop P\limits_{C \leftarrow B} \) represents?]

b. Find a basis \(\left\{ {{{\bf{w}}_{\bf{1}}},\,{{\bf{w}}_{\bf{2}}},\,{{\bf{w}}_{\bf{3}}}} \right\}\) for \({\mathbb{R}^{\bf{3}}}\) such that Pis the change of coordinate matrix from \(\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}},\,{{\bf{v}}_{\bf{3}}}} \right\}\) to \(\left\{ {{{\bf{w}}_{\bf{1}}},\,{{\bf{w}}_{\bf{2}}},\,{{\bf{w}}_{\bf{3}}}} \right\}\).

Short Answer

Expert verified

a. \(\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 6}\\{ - 5}\\{21}\end{array}} \right],\,\left[ {\begin{array}{*{20}{c}}{ - 6}\\{ - 9}\\{32}\end{array}} \right],\,\,\left[ {\begin{array}{*{20}{c}}{ - 5}\\0\\3\end{array}} \right]} \right\}\)

b. \(\left\{ {\left[ {\begin{array}{*{20}{c}}{28}\\{ - 9}\\{ - 3}\end{array}} \right],\,\left[ {\begin{array}{*{20}{c}}{38}\\{ - 13}\\2\end{array}} \right],\,\,\left[ {\begin{array}{*{20}{c}}{21}\\{ - 7}\\3\end{array}} \right]} \right\}\)

Step by step solution

01

Find the basis for \(\left\{ {{{\bf{u}}_{\bf{1}}},\,{{\bf{u}}_{\bf{2}}},\,{{\bf{u}}_{\bf{3}}}} \right\}\)

Using the definition of C coordinate vector, you get

\(\begin{aligned} \left[ {\begin{array}{*{20}{c}}{{{\bf{u}}_1}}&{{{\bf{u}}_2}}&{{{\bf{u}}_3}}\end{array}} \right] &= \left[ {\begin{array}{*{20}{c}}{{{\bf{v}}_1}}&{{{\bf{v}}_2}}&{{{\bf{v}}_3}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\bf{u}}_1}} \right]}_C}}&{{{\left[ {{{\bf{u}}_2}} \right]}_C}}&{{{\left[ {{{\bf{u}}_3}} \right]}_C}}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{{{\bf{v}}_1}}&{{{\bf{v}}_2}}&{{{\bf{v}}_3}}\end{array}} \right]P\\ &= \left[ {\begin{array}{*{20}{c}}{ - 2}&{ - 8}&{ - 7}\\2&5&2\\3&2&6\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&2&{ - 1}\\{ - 3}&{ - 5}&0\\4&6&1\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 6}&{ - 6}&{ - 5}\\{ - 5}&{ - 9}&0\\{21}&{32}&3\end{array}} \right].\end{aligned}\)

So, the basis is given by the columns \(\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 6}\\{ - 5}\\{21}\end{array}} \right],\,\left[ {\begin{array}{*{20}{c}}{ - 6}\\{ - 9}\\{32}\end{array}} \right],\,\,\left[ {\begin{array}{*{20}{c}}{ - 5}\\0\\3\end{array}} \right]} \right\}\).

02

Find the basis for \(\left\{ {{{\bf{w}}_{\bf{1}}},\,{{\bf{w}}_{\bf{2}}},\,{{\bf{w}}_{\bf{3}}}} \right\}\)

Using the definition of C coordinate vector, you get

\(\begin{aligned} \left[ {\begin{array}{*{20}{c}}{{{\bf{w}}_1}}&{{{\bf{w}}_2}}&{{{\bf{w}}_3}}\end{array}} \right] &= \left[ {\begin{array}{*{20}{c}}{ - 2}&{ - 8}&{ - 7}\\2&5&2\\3&2&6\end{array}} \right]{\left[ {\begin{array}{*{20}{c}}1&2&{ - 1}\\{ - 3}&{ - 5}&0\\4&6&1\end{array}} \right]^{ - 1}}\\ &= \left[ {\begin{array}{*{20}{c}}{ - 2}&{ - 8}&{ - 7}\\2&5&2\\3&2&6\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5&8&5\\{ - 3}&{ - 5}&{ - 3}\\{ - 2}&{ - 2}&{ - 1}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{28}&{38}&{21}\\{ - 9}&{ - 13}&{ - 7}\\{ - 3}&2&3\end{array}} \right].\end{aligned}\)

So, the basis is given by the columns \(\left\{ {\left[ {\begin{array}{*{20}{c}}{28}\\{ - 9}\\{ - 3}\end{array}} \right],\,\left[ {\begin{array}{*{20}{c}}{38}\\{ - 13}\\2\end{array}} \right],\,\,\left[ {\begin{array}{*{20}{c}}{21}\\{ - 7}\\3\end{array}} \right]} \right\}\).

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Most popular questions from this chapter

In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

\({\left( {{\bf{2}} - t} \right)^{\bf{3}}}\), \({\left( {{\bf{3}} - t} \right)^2}\), \({\bf{1}} + {\bf{6}}t - {\bf{5}}{t^{\bf{2}}} + {t^{\bf{3}}}\)

Use coordinate vector to test whether the following sets of poynomial span \({{\bf{P}}_{\bf{2}}}\). Justify your conclusions.

a. \({\bf{1}} - {\bf{3}}t + {\bf{5}}{t^{\bf{2}}}\), \( - {\bf{3}} + {\bf{5}}t - {\bf{7}}{t^{\bf{2}}}\), \( - {\bf{4}} + {\bf{5}}t - {\bf{6}}{t^{\bf{2}}}\), \({\bf{1}} - {t^{\bf{2}}}\)

b. \({\bf{5}}t + {t^{\bf{2}}}\), \({\bf{1}} - {\bf{8}}t - {\bf{2}}{t^{\bf{2}}}\), \( - {\bf{3}} + {\bf{4}}t + {\bf{2}}{t^{\bf{2}}}\), \({\bf{2}} - {\bf{3}}t\)

Suppose \({{\bf{p}}_{\bf{1}}}\), \({{\bf{p}}_{\bf{2}}}\), \({{\bf{p}}_{\bf{3}}}\), and \({{\bf{p}}_{\bf{4}}}\) are specific polynomials that span a two-dimensional subspace H of \({P_{\bf{5}}}\). Describe how one can find a basis for H by examining the four polynomials and making almost no computations.

Question 11: Let \(S\) be a finite minimal spanning set of a vector space \(V\). That is, \(S\) has the property that if a vector is removed from \(S\), then the new set will no longer span \(V\). Prove that \(S\) must be a basis for \(V\).

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

19. \(A = \left( {\begin{array}{*{20}{c}}{.9}&1&0\\0&{ - .9}&0\\0&0&{.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right)\).
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