How can one define the trace of a linear operator on any finite dimensional vector space, using the fact that $tr(A) = tr(P^{-1}AP)$?

Question

Firstly, I had to prove that $tr(AB) = tr(BA)$ and deduce that the trace is an invariant of similarity i.e. that $tr(A) = tr(P^{-1}AP)$ for any $A$ and invertible $P$. I could prove the first part - Now here's my deduction:

Suppose $B$ is similar to $A$ i.e. $B = P^{-1}AP$
Consider $tr(B) = tr(P^{-1}AP) = tr(APP^{-1}) = tr(A)$

Would this suffice?

I then have to explain how I can use this fact to define the trace of a linear operator $\phi : V \rightarrow V$ on any finite dimensional vector space $V$. The hint given says to consider the actual definition of determinant.

I'm not sure where to start - any help would be much appreciated.

Answer 1

Your argument is correct! Now consider an operator $\phi:V\to V$, chose any basis $(e_i)$, suppose $\phi$ is given by matrix $A$, define $tr(\phi):=tr(A)$. What remains is to show this do not depend on the basis.