Question

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

15.

a. If \({\mathop{\rm x}\nolimits} \) is in \(V\) and if \(B\) contains \(n\) vectors, then the \(B - \)coordinate vector of \({\mathop{\rm x}\nolimits} \) is in \({\mathbb{R}^n}\).

b. If \({P_B}\) is the change-of-coordinates matrix, then \({\left( {\mathop{\rm x}\nolimits} \right)_B} = {P_B}{\mathop{\rm x}\nolimits} \), for x in \(V\).

c. The vector spaces \({{\mathop{\rm P}\nolimits} _3}\) and \({\mathbb{R}^3}\) are isomorphic.

Short answer

a. The given statement is true.

b. The given statement is false.

c. The given statement is false.

Step-by-step solution

Determine whether the given statement is true or false

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.

Determine whether the given statement is true or false

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.

Determine whether the given statement is true or 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.