Editing Trace (linear algebra) (section)

== Trace of a linear operator ==
In general, given some linear map {{math|''f'' : ''V'' → ''V''}} (where {{mvar|V}} is a finite-[[dimension (linear algebra)|dimensional]] [[vector space]]), we can define the trace of this map by considering the trace of a [[Representation theory|matrix representation]] of {{mvar|f}}, that is, choosing a [[basis (linear algebra)|basis]] for {{mvar|V}} and describing {{mvar|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 [[matrix similarity|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 [[natural isomorphism|canonical isomorphism]] between the space {{math|End(''V'')}} of linear maps on {{mvar|V}} and {{math|''V'' ⊗ ''V''*}}, where {{math|''V''*}} is the [[dual space]] of {{mvar|V}}. Let {{mvar|v}} be in {{mvar|V}} and let {{mvar|g}} be in {{mvar|''V''*}}. Then the trace of the indecomposable element {{math|''v'' ⊗ ''g''}} is defined to be {{math|''g''(''v'')}}; the trace of a general element is defined by linearity. The trace of  a linear map {{math|''f'' : ''V'' → ''V''}} can then be defined as the trace, in the above sense, of the element of {{math|''V'' ⊗ ''V''*}} corresponding to ''f'' under the above mentioned canonical isomorphism. Using an explicit basis for {{mvar|V}} and the corresponding dual basis for {{math|''V''*}}, one can show that this gives the same definition of the trace as given above.