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

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

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 \(V\).