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Chapter 4: Q33E (page 191)

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\(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\).

Short Answer

Expert verified

The set \(\left\{ {{p_1},{p_2}} \right\}\) is a basis for \({\rm{Span}}\left\{ {{p_1},{p_2},{p_3}} \right\}\).

Step by step solution

01

Find the relation among the polynomials

\(\begin{array}{c}{p_1}\left( t \right) + {p_2}\left( t \right) = 1 + t + 1 - t\\ = 2\\{p_1}\left( t \right) + {p_2}\left( t \right) = {p_3}\left( t \right)\end{array}\)

By the spanning set theorem, you get

\({\rm{Span}}\left\{ {{p_1},{p_2},{p_3}} \right\} = {\rm{Span}}\left\{ {{p_1},{p_2}} \right\}\).

02

Use the definition of linear independence

Consider the linear combination of \({p_1}\) and \({p_2}\).

\(\begin{array}{c}{c_1}{p_1}\left( t \right) + {c_2}{p_2}\left( t \right) = 0\\{c_1}\left( {1 + t} \right) + {c_2}\left( {1 - t} \right) = 0\\{c_1} + {c_1}t + {c_2} - {c_2}t = 0\\\left( {{c_1} + {c_2}} \right) + \left( {{c_1} - {c_2}} \right)t = 0\end{array}\)

This implies,

\(\begin{array}{l}{c_1} + {c_2} = 0\\{c_1} - {c_2} = 0.\end{array}\)

Thus, \({c_1} = {c_2} = 0\).

This implies \(\left\{ {{p_1},{p_2}} \right\}\) is linearly independent.

03

Draw a conclusion

Hence, \(\left\{ {{p_1},{p_2}} \right\}\) forms a basis for \({\rm{Span}}\left\{ {{p_1},{p_2},{p_3}} \right\}\).

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Most popular questions from this chapter

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

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

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

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

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

Let V be a vector space that contains a linearly independent set \(\left\{ {{u_{\bf{1}}},{u_{\bf{2}}},{u_{\bf{3}}},{u_{\bf{4}}}} \right\}\). Describe how to construct a set of vectors \(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\) in V such that \(\left\{ {{v_{\bf{1}}},{v_{\bf{3}}}} \right\}\) is a basis for Span\(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\).

In Exercise 2, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

2. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{5}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{7}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{5}}}\end{array}} \right)\)
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