In Serge Lang’s third edition of Linear Algebra on p. 243, he gives the result of the Hamilton-Cayley theorem for linear operators for vector spaces over any field $K$. My problem is that I don’t understand at all how it’s a proof. If it’s meant to be more hint oriented, then I suppose I don’t get where to start with the hints. I completely understand his proof over the field $\mathbb{C}$ - but not this. It’s the following:
Let $V$ be a finite dimensional vector space over the field $K$, and let $A: V\to V$ be a linear map. Let $P$ be the characteristic polynomial of $A$. Then $P(A)=0$.
Proof. Take a basis of $V$, and let $M$ be the matrix representing $A$ with respect to this basis. Then $P_M=P_A$, and it suffices to prove that $P_M(M)=0$. But we can apply Theorem $2.1$ to conclude the proof.
Theorem $2.1$ refers to the previously proved result that the Hamilton-Cayley holds over $\mathbb{C}$.
I guess I don’t know how to show $P_M(M)=0$, and how $2.1$ would help after. Any help would be appreciated.
I think in Lang's text, $K$ is assumed to be a sub-field of $\mathbb{C}$ (see the very first chapter). So $M$ is a matrix of complex numbers, for which you can apply Theorem 2.1.