Proving that L(V) cannot be finite dimensional given an identity.
Solution 1:
Hint: define the linear operator $\Phi(A) = AT - TA$ on $L(V)$. Note that, on the polynomials of $S$, it works a little like differentiation. Think about how you might prove $\{1, x, x^2, \ldots, x^n\}$ are linearly independent polynomial functions using differentiation, and perhaps you might be able to adapt the proof.
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