Let \(V\) be the space of all \(n \times n\) matrices over a field \(F\) and let \(B\) be a fixed \(n \times n\) matrix. If \(T\) is the linear operator on \(V\) defined by \(T(A) = AB - BA\), and if \(f\) is the trace function, what is \(T^t f\)?
Solution
For \(A\in V\), \[ (T^tf)(A)=f(T(A))=\operatorname{tr}(AB-BA). \] Since \(\operatorname{tr}(AB)=\operatorname{tr}(BA)\), \[ (T^tf)(A)=0 \] for every \(A\). Hence \[ T^tf=0. \]