I understand the definition of trace and linear operator individually but I don't seem to understand as to what does it mean by trace of a linear operator on a finite dimensional linear space. What I have found out is that trace of a linear operator on a finite dimensional linear space is the trace of any matrix which represents the operator relative to an ordered basis of the space. I am confused as why is this definition well defined. If T:V->V is the linear operator defined on V by T(A)=BA for all A in V and B is a fixed matrix. How do I represent T relative to standard ordered basis for V where V is the linear space of all 2X2 real matrices.