Answer
See solution.
Work Step by Step
We are required to show that the coordinate mapping is one-to-one.
Suppose $[ \mathbf { u } ] _ { B } = [ \mathbf { w } ] _ { B }$
are we have the equations as;
$\begin{aligned} \mathbf { u } & = p _ { 1 } \mathbf { b } _ { 1 } + \ldots + p _ { n } \mathbf { b } _ { n } \\ \mathbf { w } & = q _ { 1 } \mathbf { b } _ { 1 } + \ldots + q _ { n } \mathbf { b } _ { n } \end{aligned}$
Then by definition of a coordinate vector:
$\begin{array}{l}\left[ {\begin{array}{*{20}{c}}{{p_1}}\\\vdots \\{{p_n}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{q_1}}\\\vdots \\{{q_n}}\end{array}} \right]\\\end{array}$
So
$p_{1}=q_{1},p_{2}=q_{2},\dots,p_{n}=q_{n}$. Therefore $\mathbf { u}= \mathbf { w}$ and the mapping $\mathbf { x}\mapsto[\mathbf { x}]_{ B}$
is one-to-one.
