Find the following (Hermitian) inner products:

(a) $\langle \mathbf{u}, \mathbf{v}\rangle,$ where $$\mathbf{u} = \begin{bmatrix} 3 \\ 2 \\ 10 \\ 3\end{bmatrix} \text{ and } \mathbf{v} = \begin{bmatrix} 6 \\ 8 \\ 6 \\8 \end{bmatrix}.$$

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

(b) $\langle \mathbf{u}, \mathbf{v}\rangle,$ where $$\mathbf{u} = \begin{bmatrix} -1 \\ -1 \\ 3 \\ -3 \\ -3 \end{bmatrix} \text{ and } \mathbf{v} = \begin{bmatrix} 6 \\ 8 \\ 6 \\6 \\ 6 \end{bmatrix}.$$

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

(c) $\langle \mathbf{u}, \mathbf{v}\rangle,$ where $$\mathbf{u} = \begin{bmatrix} 7 + 8i \\ 6 + 7i \\ 7 + 4i \end{bmatrix} \text{ and } \mathbf{v} = \begin{bmatrix} 7 + 4i \\ 5 + 6i \\ 2 + 2i \end{bmatrix}.$$

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