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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\(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\).

Short Answer

Expert verified

If \({v_1}\) and \({v_3}\) are linearly independent and \({v_2} = a{v_1} + b{v_3}\) and \({v_4} = c{v_1} + d{v_3}\), wherea, b, c, and d are scalar, then \(\left\{ {{v_1},{v_3}} \right\}\) is a basis for \({\rm{Span}}\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\).

Step by step solution

01

Construct the linearly independent vectors

Let \({v_1}\) and \({v_3}\) be linearly independent.

02

Construct a relation between the vectors, except \({v_{\bf{1}}}\) and

Let \({v_2} = a{v_1} + b{v_3}\) and \({v_4} = c{v_1} + d{v_3}\), wherea, b, c, and d are scalar.

By the spanning set theorem, \({\rm{Span}}\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\} = {\rm{Span}}\left\{ {{v_1},{v_3}} \right\}\).

03

Draw a conclusion

In this way, \(\left\{ {{v_1},{v_3}} \right\}\) forms a basis for \({\rm{Span}}\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\).

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Most popular questions from this chapter

If the null space of a\({\bf{5}} \times {\bf{6}}\)matrix A is 4-dimensional, what is the dimension of the row space of A?

Consider the polynomials \({{\bf{p}}_{\bf{1}}}\left( t \right) = {\bf{1}} + t\), \({{\bf{p}}_{\bf{2}}}\left( t \right) = {\bf{1}} - t\), \({{\bf{p}}_{\bf{3}}}\left( t \right) = {\bf{4}}\), \({{\bf{p}}_{\bf{4}}}\left( t \right) = {\bf{1}} + {t^{\bf{2}}}\), and \({{\bf{p}}_{\bf{5}}}\left( t \right) = {\bf{1}} + {\bf{2}}t + {t^{\bf{2}}}\), and let H be the subspace of \({P_{\bf{5}}}\) spanned by the set \(S = \left\{ {{{\bf{p}}_{\bf{1}}},\,{{\bf{p}}_{\bf{2}}},\;{{\bf{p}}_{\bf{3}}},\,{{\bf{p}}_{\bf{4}}},\,{{\bf{p}}_{\bf{5}}}} \right\}\). Use the method described in the proof of the Spanning Set Theorem (Section 4.3) to produce a basis for H. (Explain how to select appropriate members of S.)

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.

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

Verify that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\) if \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\5\end{array}} \right]\) and \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}} \right]\).

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