In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$. In this subsection, we place ourselves in the case where $p = n$. II.B.1) Specify the dimension of $E_i$ for all $i$ in $\llbracket 1, p \rrbracket$. II.B.2) How many lines of $E$ are stable by $f$? II.B.3) If $n \geqslant 3$ and $k \in \llbracket 2, n-1 \rrbracket$, how many subspaces of $E$ of dimension $k$ and stable by $f$ are there? II.B.4) How many subspaces of $E$ are stable by $f$ in this case? Give them all.
In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$. In this subsection, we place ourselves in the case where $p = n$.
\textbf{II.B.1)} Specify the dimension of $E_i$ for all $i$ in $\llbracket 1, p \rrbracket$.
\textbf{II.B.2)} How many lines of $E$ are stable by $f$?
\textbf{II.B.3)} If $n \geqslant 3$ and $k \in \llbracket 2, n-1 \rrbracket$, how many subspaces of $E$ of dimension $k$ and stable by $f$ are there?
\textbf{II.B.4)} How many subspaces of $E$ are stable by $f$ in this case? Give them all.