Define $f: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ by

$$f(a, b, c)=(a+3 b-c, 2 b+c,-4 b-c)$$

Find the characteristic polynomial and the minimal polynomial of $f$. Is $f$ diagonalisable? Are $f$ and $f^{2}$ linearly independent endomorphisms of $\mathbb{C}^{3}$ ? Justify your answers.