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Chapter 4: Q3E (page 191)

Let \(u = \left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}} \right\}\) and \(W = \left\{ {{{\mathop{\rm w}\nolimits} _1},{{\mathop{\rm w}\nolimits} _2}} \right\}\) be bases for \(V\), and let \(P\) be a matrix whose columns are \({\left[ {{{\mathop{\rm u}\nolimits} _1}} \right]_w}\) and \({\left[ {{{\mathop{\rm u}\nolimits} _2}} \right]_w}\). Which of the following equations is satisfied by \(P\) for all x in \(V\)?

  1. \({\left[ {\mathop{\rm x}\nolimits} \right]_u} = P{\left[ {\mathop{\rm x}\nolimits} \right]_w}\)
  2. \({\left[ {\mathop{\rm x}\nolimits} \right]_w} = P{\left[ {\mathop{\rm x}\nolimits} \right]_u}\)

Short Answer

Expert verified

Equation (ii) is satisfied by \(P\) for all \({\mathop{\rm x}\nolimits} \) in \(V\).

Step by step solution

01

Change-of-coordinate matrix

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 according to Theorem 15, 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 \(B\). That is, \(\mathop P\limits_{C \leftarrow B} = \left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\mathop{\rm b}\nolimits} _1}} \right]}_C}}&{{{\left[ {{{\mathop{\rm b}\nolimits} _2}} \right]}_C}}& \cdots &{{{\left[ {{{\mathop{\rm b}\nolimits} _n}} \right]}_C}}\end{array}} \right]\).

02

Determine which equation is satisfied by \(P\)

Let \(P\) be the matrix whose columns are \({\left[ {{{\mathop{\rm u}\nolimits} _1}} \right]_w}\) and \({\left[ {{{\mathop{\rm u}\nolimits} _2}} \right]_w}\).

Since the columns of \(P\) are \(w - \)coordinate vectors, a vector of the form \(P{\mathop{\rm x}\nolimits} \) must be a \(w - \)coordinate vector. Thus \(P\) satisfies equation (ii).

Thus, equation (ii) is satisfied by \(P\) for all \({\mathop{\rm x}\nolimits} \) in \(V\).

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Most popular questions from this chapter

In Exercise 3, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

3. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{2}}\\{ - {\bf{2}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{0}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{0}}\\{ - {\bf{1}}}\end{array}} \right)\)

If A is a \({\bf{7}} \times {\bf{5}}\) matrix, what is the largest possible rank of A? If Ais a \({\bf{5}} \times {\bf{7}}\) matrix, what is the largest possible rank of A? Explain your answer.

Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.

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