Moving one factor from the front to the back preserves trace when products are defined. The point is to make the formal expression readable enough to audit line by line.
Start locally
Tensor and exterior powers turn multilinear behavior into linear maps. For finite-dimensional \(V\), the spaces \(V^{\otimes k}\) and \(\bigwedge^kV\) carry induced actions of every \(T\in\operatorname{End}(V)\).
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
Compute before generalising
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
The global view
A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.
Edge conditions
Tensor coordinates depend on a basis even when the tensor does not. Index notation is safe only when contraction rules and variance are clear.
The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.