Past exam of the linear algebra course of the University of Oxford / 2017 / 2 / a / iii / Solution

For $\varphi \in V^{\prime }$ and $v\in V$,so $I^{\prime }$ is the identity map on the dual space. If $S:U\to V$ and $T:V\to W$, then, for $\psi \in W^{\prime }$ and $u\in U$,Equality on all vectors provesApply this to $R{R}^{-1}=R^{-1}R=I$. It gives both $R^{\prime }(R^{-1}{)}^{\prime }=I^{\prime }$ and $(R^{-1}{)}^{\prime }{R}^{\prime }=I^{\prime }$. Consequently the dual map is invertible and