Trace (linear Algebra) - Trace of A Linear Operator

Trace of A Linear Operator

Given some linear map f : V → V (V is a finite-dimensional vector space) generally, we 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. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map.

Such a definition can be given using the canonical isomorphism between the space End(V) of linear maps on V and V⊗V*, where V* is the dual space of V. Let v be in V and let f be in V*. Then the trace of the decomposable element v⊗f is defined to be f(v); the trace of a general element is defined by linearity. Using an explicit basis for V and the corresponding dual basis for V*, one can show that this gives the same definition of the trace as given above.