I have a question regarding the trace of a linear map $f: V \to V$. As usual, one can define the trace of this map by considering the trace of matrix representation of $f$, that is, choosing a basis for $V$ and describing $f$ as a matrix relative to this basis, and taking the trace of this square matrix.
What I don't get, and is something appearing in my book, is how comes that the trace is then equal to:
where $\langle \;\_ \;, \;\_ \;\rangle$ stands for an inner product and $e_i$'s form an orthonormal basis.
Thanks in advance for your help.