Consider the inner product $\langle X,Y \rangle:= \text{trace}(X^T Y)$ on the vector space $V=\mathbb{R}^{2\times 2}$. The linear operator $L:V \to V$ is defined by $L(X)= \left[\begin{array}{cc} -5 &1\cr -5 &5 \end{array}\right] X$. Find the adjoint $L^*$ of $L$.
$L^*(\left[\matrix {a & b \\ c & d}\right]) =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$