Space of matrices/Standard basis/Dual basis/Trace/Exercise

Let ${}V=\operatorname {Mat} _{n}(K)$ be the space of the ${}n\times n$-matrices over the field ${}K$, with the standard basis ${}e_{ij}$. Describe the trace as a linear combination with respect to the dual basis ${}e_{ij}^{*}$.

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