Dual mapping/Dual basis/Matrix/Fact/Proof

The claim means the equality

${\displaystyle {}\varphi ^{*}(w_{j}^{*})=\sum _{i=1}^{n}a_{ji}v_{i}^{*}\,}$

in ${\displaystyle {}{V}^{*}}$. This can be checked on the basis ${\displaystyle {}v_{k}}$, ${\displaystyle {}k=1,\ldots ,n}$. On one hand, we have

${\displaystyle {}{\begin{aligned}{\left(\varphi ^{*}(w_{j}^{*})\right)}(v_{k})&=w_{j}^{*}{\left(\varphi (v_{k})\right)}\\&=w_{j}^{*}{\left(\sum _{r=1}^{m}a_{rk}w_{r}\right)}\\&=\sum _{r=1}^{m}a_{rk}w_{j}^{*}(w_{r})\\&=a_{jk},\end{aligned}}}$

on the other hand, we have

${\displaystyle {}{\left(\sum _{i=1}^{n}a_{ji}v_{i}^{*}\right)}(v_{k})=a_{jk}\,.}$