Linear Functional Trace

Question

Let $V$ be the vector space of $n \times n$ matrices over the field $\mathbf{F}$.

How to show that every linear functional on V can be written as $f_B(A) = \operatorname{trace}(AB)$.

Answer 1

Sketch of Proof: Note that $V$ with $\langle B, A\rangle = \operatorname{Tr}(B^\ast A)$ is a $n^2$-dimensional inner product space over $\mathbb{C}$.

In particular, we can see that the set of matrices $\{e_{ij}\}_{i=1, j=1}^{n}$ where \begin{align} (e_{ij})_{k\ell} = \delta_{ik}\delta_{j\ell} \end{align} is an orthongal basis, then for every linear functional $T:V\rightarrow \mathbb{C}$ we have that \begin{align} T(A) = \sum^{n}_{i=1}\sum^n_{j=1}a_{ij} T(e_{ij}) = \sum^{n}_{i=1}\sum^n_{j=1}a_{ij} b_{ij} = \operatorname{Tr}(B A) \end{align} where $A= \sum^n_{i=1}\sum^n_{j=1} a_{ij}e_{ij}$ and $B= \sum^n_{i=1}\sum^n_{j=1}b_{ij}e_{ij}=\sum^n_{i=1}\sum^n_{j=1}T(e_{ij})e_{ij}$.