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Chapter 4: Q10Q (page 191)

In Exercises 9 and 10, find the change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^n}\).

10. \(B = \left\{ {\left[ {\begin{array}{*{20}{c}}3\\{ - 1}\\4\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\0\\{ - 5}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}8\\{ - 2}\\7\end{array}} \right]} \right\}\)

Short Answer

Expert verified

The change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^3}\) is \({P_B} = \left[ {\begin{array}{*{20}{c}}3&2&8\\{ - 1}&0&{ - 2}\\4&{ - 5}&7\end{array}} \right]\).

Step by step solution

01

State the change of coordinates

Let\({P_B} = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}& \cdots &{{{\mathop{\rm b}\nolimits} _n}}\end{array}} \right]\), then thevector equation\[{\mathop{\rm x}\nolimits} = {c_1}{{\mathop{\rm b}\nolimits} _1} + {c_2}{{\mathop{\rm b}\nolimits} _2} + ... + {c_n}{{\mathop{\rm b}\nolimits} _n}\] becomes equivalent to \({\mathop{\rm x}\nolimits} = {P_B}{\left[ {\mathop{\rm x}\nolimits} \right]_B}\). \({P_B}\) denotes thechange-of-coordinates matrixfrom \(B\) to the standard basis in \({\mathbb{R}^n}\).

02

Determine the change-of-coordinates matrix from \(B\)

The change-of-coordinates matrix from\(B\)to the standard basis in\({\mathbb{R}^3}\)is shown below.

\[\begin{array}{c}{P_B} = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}&{{{\mathop{\rm b}\nolimits} _3}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}3&2&8\\{ - 1}&0&{ - 2}\\4&{ - 5}&7\end{array}} \right]\end{array}\]

Thus, the change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^3}\) is \({P_B} = \left[ {\begin{array}{*{20}{c}}3&2&8\\{ - 1}&0&{ - 2}\\4&{ - 5}&7\end{array}} \right]\).

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Most popular questions from this chapter

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

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

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

21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.

Consider the polynomials , and \({p_{\bf{3}}}\left( t \right) = {\bf{2}}\) \({p_{\bf{1}}}\left( t \right) = {\bf{1}} + t,{p_{\bf{2}}}\left( t \right) = {\bf{1}} - t\)(for all t). By inspection, write a linear dependence relation among \({p_{\bf{1}}},{p_{\bf{2}}},\) and \({p_{\bf{3}}}\). Then find a basis for Span\(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\).

Explain what is wrong with the following discussion: Let \({\bf{f}}\left( t \right) = {\bf{3}} + t\) and \({\bf{g}}\left( t \right) = {\bf{3}}t + {t^{\bf{2}}}\), and note that \({\bf{g}}\left( t \right) = t{\bf{f}}\left( t \right)\). Then, \(\left\{ {{\bf{f}},{\bf{g}}} \right\}\) is linearly dependent because g is a multiple of f.
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