Let $V$ be a $n$-dimensional unitary vector space with $n < \infty$. Let $(e_1, \ldots, e_n)$ be an ON-basis of $V$ with dual basis $(e_1^*, \ldots, e_n^*)$. Define a sesquilinear form in $V \otimes V^*$ with
$\langle e_i \otimes e_j^*, e_k \otimes e_l^* \rangle _{V \otimes V^*} := \delta _{ik}\, \delta _{jl}$
Define a vector space isomorphism with
$\Phi _V \colon M(n \times n; \mathbb {C}) \to V \otimes V^*\,, \quad T \mapsto \sum _{i,j = 1}^n T_{ij}\, e_i \otimes e_j^*\,.$
Which of the following identities is true for all $S,T \in M(n \times n; \mathbb {C})$?
a. $\langle \Phi _V(S), \Phi _V(T) \rangle _{V \otimes V^*} = \mathrm {tr}(\overline {T} S)$
b. $\langle \Phi _V(S), \Phi _V(T) \rangle _{V \otimes V^*} = \mathrm {tr}(T^t S)$
c. $\langle \Phi _V(S), \Phi _V(T) \rangle _{V \otimes V^*} = \mathrm {tr}(\overline {T}^t S)$
Best Answer
Choice c is correct, assuming that $\bar T^t$ refers to the conjugate of the transpose of $T$. To see that this is the case, it suffices to note that the form is conjugate linear in the second argument and to plug in $T = v_i v_j^T$, where $v_1,\dots,v_n$ denotes the canonical basis of $\Bbb C^{n}$.