Let $$\mathbf{u} = \begin{bmatrix} 2 + 6i \\ 8 + 5i \\ 4 + 6i \\ 7 + 6i \\ 2 + 5i \\ 7 + 4i \end{bmatrix} \text{ and } \mathbf{v} = \begin{bmatrix} 2 + 7i \\ 2 + 7i \\ 4 + 7i \\ 2 + 5i \\ 7 + 8i \\ 6 + 7i \end{bmatrix}.$$ Find each of the following products:
$\overline{ 2 + 6i} ( 2 + 7i ) =$
$\overline{ 8 + 5i} ( 2 + 7i ) =$
$\overline{ 4 + 6i} ( 4 + 7i ) =$
$\overline{ 7 + 6i} ( 2 + 5i ) =$
$\overline{ 2 + 5i} ( 7 + 8i ) =$
$\overline{ 7 + 4i} ( 6 + 7i ) =$

Now, find the (Hermitian) inner product $\langle \mathbf{u}, \mathbf{v}\rangle$:

Answer: $\langle \mathbf{u}, \mathbf{v}\rangle =$