R^2/Adjoint endomorphism/Explicit inner product/2/Exercise

We consider the linear mapping $\phi :{ℝ}^2\to {ℝ}^2$ given by the matrix $\left(\begin{array}{cc}-6& -3\\ 4& 5\end{array}\right)$ with respect to the standard basis. We equip ${ℝ}^2$ with the inner product $\mathrm{\Psi }$ given by $\mathrm{\Psi }(e_1,e_1)=3$, $\mathrm{\Psi }(e_2,e_2)=7$, and $\mathrm{\Psi }(e_1,e_2)=2$. Determine the matrix of the adjoint endomorphism of $\phi$ with respect to the given inner product, and with respect to the basis $\left(\begin{array}{c}4\\ -3\end{array}\right),\left(\begin{array}{c}-2\\ -7\end{array}\right)$.

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