Let \(V\) be a finite-dimensional vector space over the field \(F\). Show that \(T \to T^t\) is an isomorphism of \(L(V, V)\) onto \(L(V^*, V^*)\).
Solution
Choose an ordered basis \(\mathcal B\) of \(V\) and let \(\mathcal B^*\) be its dual basis. If \(A=[T]_{\mathcal B}\), then \[ [T^t]_{\mathcal B^*}=A^t. \] Thus, under the standard identifications \[ L(V,V)\cong M_n(F),\qquad L(V^*,V^*)\cong M_n(F), \] the map \[ T\longmapsto T^t \] corresponds exactly to matrix transposition \[ A\longmapsto A^t. \] Matrix transposition is linear and satisfies \((A^t)^t=A\), so it is an isomorphism. Therefore \(T\mapsto T^t\) is an isomorphism of \(L(V,V)\) onto \(L(V^*,V^*)\).