I am searching for a short coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ for linear operators $A$, $B$ between finite dimensional vector spaces of the same dimension.
The usual proof is to represent the operators as matrices and then use matrix multiplication. I want a coordinate-free proof. That is, one that does not make reference to an explicit matrix representation of the operator. I define trace as the sum of the eigenvalues of an operator.
Ideally, the proof the should be shorter and require fewer preliminary lemmas than the one given in this blog post.
I would be especially interested in a proof that generalizes to the trace class of operators on a Hilbert space.
Best Answer
$\newcommand{\tr}{\operatorname{tr}}$Here is an exterior algebra approach. Let $V$ be an $n$-dimensional vector space and let $\tau$ be a linear operator on $V$. The alternating multilinear map $$ (v_1,\dots,v_n) \mapsto \sum_{k=1}^n v_1 \wedge\cdots\wedge \tau v_k \wedge\cdots\wedge v_n $$ induces a unique linear operator $\psi: \bigwedge^n V \to \bigwedge^n V$. The trace $\tr(\tau)$ is defined as the unique number satisfying $\psi = \tr(\tau)\iota$, where $\iota$ is the identity. (This is possible because $\bigwedge^n V$ is one-dimensional.)
Let $\sigma$ be another linear operator. We compute \begin{align} (\tr\sigma)(\tr\tau) v_1 \wedge\cdots\wedge v_n &= \sum_{k=1}^n (\tr\sigma) v_1 \wedge\cdots\wedge \tau v_k \wedge\cdots\wedge v_n \\ &= \sum_{k=1}^n v_1 \wedge\cdots\wedge \sigma \tau v_k \wedge\cdots\wedge v_n \\ & \qquad + \sum_{k=1}^n \sum_{j \ne k} v_1 \wedge\cdots\wedge \sigma v_j \wedge \cdots \wedge \tau v_k \wedge\cdots\wedge v_n. \end{align}
Notice that the last sum is symmetric in $\sigma$ and $\tau$, and so is $(\tr\sigma)(\tr\tau) v_1 \wedge\cdots\wedge v_n$. Therefore $$ \sum_{k=1}^n v_1 \wedge\cdots\wedge \sigma \tau v_k \wedge\cdots\wedge v_n = \sum_{k=1}^n v_1 \wedge\cdots\wedge \tau \sigma v_k \wedge\cdots\wedge v_n, $$ i.e. $\tr(\sigma\tau)=\tr(\tau\sigma)$.
EDIT: To see that the trace is the sum of all eigenvalues, plug in your eigenvectors in the multilinear map defined at the beginning.