Let $T:V\to V$ be a linear transformation from the $n$-dimensional vector space $V$ to itself. The trace of $T$ is \(\text{tr}(T) = \sum_{i=1}^n \langle T(e_i), e_i\rangle\) where ${e_i}_{i=1}^n$ is an orthonormal basis for $V$.
Equivalently, if $V^$ is the dual ov $V$, then the trace of a linear transformation $T: V\to V$ is the functional \(\text{tr}:V^*\otimes V \to \mathbb R\) from the tensor product of of $V^$ and $V$ determined by $\phi \otimes v \mapsto \phi(v)$ for $\phi \in V^*$ and $v \in V$.
Wikidata ID: Q321102