TITLE: Proving the trace of a transformation is independent of the basis chosen
QUESTION [44 upvotes]: How would you prove the trace of a transformation from V to V (where V is finite dimensional) is independent of the basis chosen?

REPLY [9 votes]: Let $A,B$ be $n \times n$ matrices, it applies $tr(AB) = tr(BA)$.
proof $\: tr(AB) = \sum_{i} (AB)_{ii} = \sum_{i}(\sum_{k}A_{ik}B_{ki})= \sum_{k}\sum_{i} (B_{ki} A_{ik}) = \sum_{k} (BA)_{kk} = tr(BA)$ q.e.d.
Let $A$ be a $n \times n$ matrix. The change of basis is given by $U^{-1}AU$, where $U$ is an invertible $n \times n$ matrix.
We finally can see that: $\:$ $tr(U^{-1}AU) = tr(AUU^{-1}) = tr(A)$.