91Ó°ÊÓ

a)

Suppose \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) is a basis for \(V\) and x is in \(V\). Thecoordinatesof \({\mathop{\rm x}\nolimits} \) relative to basis \(B\)(or the \(B\)-coordinates of x) are the weights \({c_1},...,{c_n}\), such that \({\mathop{\rm x}\nolimits} = {c_1}{b_1} + ... + {c_n}{b_n}\).

Thus, statement (a) is true.

b)

As \({P_B} = \left( {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}& \cdots &{{{\mathop{\rm b}\nolimits} _n}}\end{array}} \right)\), thevector equation\({\mathop{\rm x}\nolimits} = {c_1}{{\mathop{\rm b}\nolimits} _1} + {c_2}{{\mathop{\rm b}\nolimits} _2} + ... + {c_n}{{\mathop{\rm b}\nolimits} _n}\)is equivalent to \({\mathop{\rm x}\nolimits} = {P_B}{\left( {\mathop{\rm x}\nolimits} \right)_B}\). \({P_B}\) represents the change-of-coordinates matrixfrom \(B\) to the standard basis in \({\mathbb{R}^n}\).

Thus, statement (b) is false.

c)

The coordinate mapping \({\mathop{\rm p}\nolimits} \mapsto {\left( {\mathop{\rm p}\nolimits} \right)_B}\) is isomorphismfrom \({{\mathop{\rm P}\nolimits} _3}\) onto \({\mathbb{R}^4}\). All vector space operations in \({{\mathop{\rm P}\nolimits} _3}\) correspond to operations in \({\mathbb{R}^4}\). \({{\mathop{\rm P}\nolimits} _3}\) is isomorphic to \({\mathbb{R}^4}\).

Thus, statement (c) is false.