If anyone could help with this..

In space $\displaystyle M_2(\mathbb{C})$ there is a set:

$\displaystyle L= \begin{Bmatrix} \begin{bmatrix} -1 & 0 \\ 0 & i \end{bmatrix} , \begin{bmatrix} 0 & i \\ 1 & 0 \end{bmatrix} \end{Bmatrix}$

Find a basis for the annihilator $\displaystyle L^0$ of $\displaystyle L$ and complete it to a basis for the dual space of $\displaystyle M_2(\mathbb{C})$

Thank you