Let \(V\) and \(W\) be vector spaces over the field \(F\) and let \(U\) be an isomorphism of \(V\) onto \(W\). Prove that \(T \rightarrow UTU^{-1}\) is an isomorphism of \(L(V, V)\) onto \(L(W, W)\).
Solution
Define \[ \Phi:L(V,V)\longrightarrow L(W,W),\qquad \Phi(T)=UTU^{-1}. \] For \(S,T\in L(V,V)\) and \(a,b\in F\), \[ \Phi(aS+bT)=U(aS+bT)U^{-1}=a\Phi(S)+b\Phi(T), \] so \(\Phi\) is linear.
Define \[ \Psi:L(W,W)\longrightarrow L(V,V),\qquad \Psi(R)=U^{-1}RU. \] Then \[ \Psi(\Phi(T))=U^{-1}(UTU^{-1})U=T \] and \[ \Phi(\Psi(R))=U(U^{-1}RU)U^{-1}=R. \] Thus \(\Psi=\Phi^{-1}\), so \(\Phi\) is an isomorphism. In fact it also preserves composition: \[ \Phi(ST)=\Phi(S)\Phi(T). \]