We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
We denote $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$, and $f_1, f_2, f_3, f_4, f_5, f_6, f_7$ the column vectors of the matrix $C$. We denote $F$ the vector subspace of $\mathbb{R}^7$ spanned by the first three column vectors $f_1, f_2$ and $f_3$ of $C$.
I.B.1) Show that $F$ is stable under $c$. I.B.2) Show that $(f_1, f_2, f_3)$ is a basis of $F$, and calculate the matrix $\Phi$ in this basis of the endomorphism $\varphi$ of $F$ induced by $c$.