Question: Determine whether the operation defines an inner product. (a) \(\langle\mathbf{u}, \mathbf{v}\rangle=\sqrt{u_{1} v_{1}+u_{2} v_{2}+u_{3} v_{3}}\) where \(\mathbf{u}=\left[\begin{array}{l}u_{1} \\ u_{2} \\ u_{3}\end{array}\right], \mathbf{v}=\left[\begin{array}{l}v_{1} \\ v_{2} \\ v_{3}\end{array}\right] \in \mathbb{R}^{3}\)

(b) \(\langle\mathbf{u}, \mathbf{v}\rangle=A \mathbf{u} \cdot A \mathbf{v}\) for any \(n \times n\) matrix \(A\) over \(\mathbb{R}^{n}\)

(c) \(\langle A, B\rangle=\operatorname{det}(A) \cdot \operatorname{det}(B)\) over \(\mathbb{R}_{2 \times 2}\)

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