I am reading "Linear Algebra Done Right Fourth Edition" by Sheldon Axler.
The following theorem is on p.144 in this book.
5.22 Suppose $V$ is finite-dimensional and $T\in\mathcal{L}(V)$. Then there is a unique monic polynomial $p\in\mathcal{P}(\mathbb{F})$ of smallest degree such that $p(T)=0$. Furthermore, $\deg p\leq\dim V$.
The author's proof of the existence part of this theorem is long and very difficult.
Surprisingly this great book is free. (https://linear.axler.net/)
The following theorem is on p.87 in this book.
3.72 Suppose $V$ and $W$ are finite-dimensional. Then $\mathcal{L}(V,W)$ is finite-dimensional and $$\dim\mathcal{L}(V,W)=(\dim V)(\dim W).$$
If we use Theorem 3.72,, I think the existence part of Theorem 5.22 is straightforward because $I,T,\dots,T^{\dim\mathcal{L}(V)}$ is linearly dependent.
The author, however, did not adopt this method.
So I am afraid if my above proof is wrong.
