Linear mapping/To finite-dimensional/Representation with linear forms/Fact/Proof

Proof

Let ${}w_{1},\ldots ,w_{n}$ be a basis of ${}W$ and ${}w_{1}^{*},\ldots ,w_{n}^{*}$ the corresponding dual basis. We set

${}f_{i}=\varphi ^{*}(w_{i}^{*})=w_{i}^{*}\circ \varphi \,.$

Then, for every vector ${}v\in V$, we have

${}{\begin{aligned}{\left(\sum _{i=1}^{n}f_{i}w_{i}\right)}(v)&=\sum _{i=1}^{n}f_{i}(v)w_{i}\\&=\sum _{i=1}^{n}{\left(w_{i}^{*}\circ \varphi \right)}(v)w_{i}\\&=\sum _{i=1}^{n}w_{i}^{*}(\varphi (v))w_{i}\\&=\varphi (v),\end{aligned}}$

where the last equation rests on fact.

To fact