Question

Let \(A = \left\{ {{{\mathop{\rm a}\nolimits} _1},{{\mathop{\rm a}\nolimits} _2},{{\mathop{\rm a}\nolimits} _3}} \right\}\) and \(D = \left\{ {{{\mathop{\rm d}\nolimits} _1},{{\mathop{\rm d}\nolimits} _2},{{\mathop{\rm d}\nolimits} _3}} \right\}\) be bases for \(V\), and let \(P = \left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\mathop{\rm d}\nolimits} _1}} \right]}_A}}&{{{\left[ {{{\mathop{\rm d}\nolimits} _2}} \right]}_A}}&{{{\left[ {{{\mathop{\rm d}\nolimits} _3}} \right]}_A}}\end{array}} \right]\). Which of the following equations is satisfied by \(P\) for all x in \(V\)?

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

Short answer

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

Step-by-step solution

State the 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 bases of a vector space \(V\). Then according toTheorem 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, \(\mathop P\limits_{C \leftarrow B} = \left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\mathop{\rm b}\nolimits} _1}} \right]}_C}}&{{{\left[ {{{\mathop{\rm b}\nolimits} _2}} \right]}_C}}& \cdots &{{{\left[ {{{\mathop{\rm b}\nolimits} _n}} \right]}_C}}\end{array}} \right]\).

Determine which equation is satisfied by \(P\)

Let \(P = \left[ {\begin{array}{*{20}{c}}{{{\left[ {{{\mathop{\rm d}\nolimits} _1}} \right]}_A}}&{{{\left[ {{{\mathop{\rm d}\nolimits} _2}} \right]}_A}}&{{{\left[ {{{\mathop{\rm d}\nolimits} _3}} \right]}_A}}\end{array}} \right]\).

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

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