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Let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},{{\mathop{\rm b}\nolimits} _2}} \right\}\) and \(C = \left\{ {{{\mathop{\rm c}\nolimits} _1},{{\mathop{\rm c}\nolimits} _2}} \right\}\) be bases for a vector space \(V\), and suppose \({{\mathop{\rm b}\nolimits} _1} = - {{\mathop{\rm c}\nolimits} _1} + 4{{\mathop{\rm c}\nolimits} _2}\) and \({{\mathop{\rm b}\nolimits} _2} = 5{{\mathop{\rm c}\nolimits} _1} - 3{{\mathop{\rm c}\nolimits} _2}\).

a. Find the change-of-coordinates matrix from \(B\) to \(C\).

b. Find \({\left[ {\mathop{\rm x}\nolimits} \right]_c}\) for \({\mathop{\rm x}\nolimits} = 5{{\mathop{\rm b}\nolimits} _1} + 3{{\mathop{\rm b}\nolimits} _2}\). Use part (a).

Short Answer

Expert verified
  1. The change-of-coordinates matrix is \(\mathop P\limits_{C \leftarrow B} = \left[ {\begin{array}{*{20}{c}}{ - 1}&5\\4&{ - 3}\end{array}} \right]\).
  2. The \(C - \)coordinate vector is \({\left[ {\mathop{\rm x}\nolimits} \right]_C} = \left[ {\begin{array}{*{20}{c}}{10}\\{11}\end{array}} \right]\).

Step by step solution

01

Change-of-coordinate matrix

Theorem 15states that let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) and \(C = \left\{ {{{\mathop{\rm c}\nolimits} _1},...,{{\mathop{\rm c}\nolimits} _n}} \right\}\) be the bases of a vector space \(V\). Then, there is a unique \(n \times n\) matrix \(\mathop P\limits_{C \leftarrow B} \) such that \({\left[ {\mathop{\rm x}\nolimits} \right]_C} = \mathop P\limits_{C \leftarrow B} {\left[ {\mathop{\rm x}\nolimits} \right]_B}\).

The columns of \(\mathop P\limits_{C \leftarrow B} \) are the \(C - \)coordinate vectors of the vectors in the basis \(B\). Thus, \(\mathop P\limits_{C \leftarrow B} = \left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\mathop{\rm b}\nolimits} _1}} \right]}_C}}&{{{\left[ {{{\mathop{\rm b}\nolimits} _2}} \right]}_C}}& \cdots &{{{\left[ {{{\mathop{\rm b}\nolimits} _n}} \right]}_C}}\end{array}} \right]\).

02

Determine the change-of-coordinate matrix from \(B\) to \(C\)

a)

Suppose \({{\mathop{\rm b}\nolimits} _1} = - {{\mathop{\rm c}\nolimits} _1} + 4{{\mathop{\rm c}\nolimits} _2}\) and \({{\mathop{\rm b}\nolimits} _2} = 5{{\mathop{\rm c}\nolimits} _1} - 3{{\mathop{\rm c}\nolimits} _2}\). Therefore, \({\left[ {{{\mathop{\rm b}\nolimits} _1}} \right]_C} = \left[ {\begin{array}{*{20}{c}}{ - 1}\\4\end{array}} \right],{\left[ {{{\mathop{\rm b}\nolimits} _2}} \right]_C} = \left[ {\begin{array}{*{20}{c}}5\\{ - 3}\end{array}} \right]\).

\(\begin{aligned} \mathop P\limits_{C \leftarrow B} &= \left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\mathop{\rm b}\nolimits} _1}} \right]}_C}}&{{{\left[ {{{\mathop{\rm b}\nolimits} _2}} \right]}_C}}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 1}&5\\4&{ - 3}\end{array}} \right]\end{aligned}\)

Thus, the change-of-coordinates matrix from \(B\) to \(C\) is \(\mathop P\limits_{C \leftarrow B} = \left[ {\begin{array}{*{20}{c}}{ - 1}&5\\4&{ - 3}\end{array}} \right]\).

03

Determine \({\left[ {\mathop{\rm x}\nolimits}  \right]_c}\) for \({\mathop{\rm x}\nolimits}  = 5{{\mathop{\rm b}\nolimits} _1} + 3{{\mathop{\rm b}\nolimits} _2}\)

b)

Suppose \({\mathop{\rm x}\nolimits} = 5{{\mathop{\rm b}\nolimits} _1} + 3{{\mathop{\rm b}\nolimits} _2}\), then \({\left[ {\mathop{\rm x}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}5\\3\end{array}} \right]\).

\(\begin{aligned} {\left[ {\mathop{\rm x}\nolimits} \right]_C} &= \mathop P\limits_{C \leftarrow B} {\left[ {\mathop{\rm x}\nolimits} \right]_B}\\ &= \left[ {\begin{array}{*{20}{c}}{ - 1}&5\\4&{ - 3}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\3\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 5 + 15}\\{20 - 9}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{10}\\{11}\end{array}} \right]\end{aligned}\)

Therefore, the \(C - \)coordinate vector is \({\left[ {\mathop{\rm x}\nolimits} \right]_C} = \left[ {\begin{array}{*{20}{c}}{10}\\{11}\end{array}} \right]\).

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Most popular questions from this chapter

Which of the subspaces \({\rm{Row }}A\) , \({\rm{Col }}A\), \({\rm{Nul }}A\), \({\rm{Row}}\,{A^T}\) , \({\rm{Col}}\,{A^T}\) , and \({\rm{Nul}}\,{A^T}\) are in \({\mathbb{R}^m}\) and which are in \({\mathbb{R}^n}\) ? How many distinct subspaces are in this list?.

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.

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{2}}}\)on the line\(y = {\bf{5}}x\).

(M) Let \({{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _5}\) denote the columns of the matrix \(A\), where \(A = \left( {\begin{array}{*{20}{c}}5&1&2&2&0\\3&3&2&{ - 1}&{ - 12}\\8&4&4&{ - 5}&{12}\\2&1&1&0&{ - 2}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}{{{\mathop{\rm a}\nolimits} _1}}&{{{\mathop{\rm a}\nolimits} _2}}&{{{\mathop{\rm a}\nolimits} _4}}\end{array}} \right)\)

  1. Explain why \({{\mathop{\rm a}\nolimits} _3}\) and \({{\mathop{\rm a}\nolimits} _5}\) are in the column space of \(B\).
  2. Find a set of vectors that spans \({\mathop{\rm Nul}\nolimits} A\).
  3. Let \(T:{\mathbb{R}^5} \to {\mathbb{R}^4}\) be defined by \(T\left( x \right) = A{\mathop{\rm x}\nolimits} \). Explain why \(T\) is neither one-to-one nor onto.

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 \(V\).

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