Proving Uniqueness of Trace Function on n X n Matrices

[guroten]

Homework Statement

Show that the trace functional on n X n matrices is unique in the following
sense. If W is the space of n X n matrices over the field F and if f is a linear functional
on W such that f(AB) = f(BA) for each A and B in W, then f is a scalar
multiple of the trace function. If, in addition, f(I) = n, then f is the trace function.

The Attempt at a Solution

I'm not sure how to start with this proof. Any help would be appreciated.

 

[Dick]

Any linear functional over nxn matrices has the form W(A)=C_{ij}*A_{ij} summed over i and j, right? Now work over a basis of the nxn matrices. Define D_{ij} to be the matrix with a 1 in the position {ij} and zero everywhere else. Now I'll just give you couple of examples. Let's just use 2x2 matrices. Let A=D_{12} and B=D_{11}. Then D_{12}*D_{11}=0 so W(D_{12}*D_{11})=0. D_{11}*D_{12}=D_{12}. So W(D_{11}*D_{12})=W(D_{12})=C_{12}. Hence C_{12}=0. Now look at D_{12} and D_{21}. Can you show C_{11}=C_{22}? Do you see how this is working?