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 $(e_1, e_2, e_3, e_4, e_5, e_6, e_7)$ the canonical basis of $\mathbb{R}^7$, and $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$. We denote $f_1, f_2, f_3, f_4, f_5, f_6, f_7$ the column vectors of the matrix $C$.
Determine a basis of the kernel and a basis of the image of $c$, as well as the rank of $c$.