91Ó°ÊÓ

We are given the orthonormal basis \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) for \(\mathbb{C}^{2}\), and the vector \(\mathbf{z}\) is expressed as follows: \[ \mathbf{z} = (4 + 2i)\mathbf{u}_1 + (6 - 5i)\mathbf{u}_2 \] We need to find the values of: \[ \mathbf{u}_{1}^{H} \mathbf{z}, \mathbf{z}^{H} \mathbf{u}_{1}, \mathbf{u}_{2}^{H} \mathbf{z}, \text{ and } \mathbf{z}^{H} \mathbf{u}_{2} \] and also determine the value of \(\|\mathbf{z}\|\).

Since \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) is an orthonormal basis, we have the following properties: \[ \mathbf{u}_1^{H}\mathbf{u}_1 = \mathbf{u}_2^{H}\mathbf{u}_2 = 1 \\ \mathbf{u}_1^{H}\mathbf{u}_2 = \mathbf{u}_2^{H}\mathbf{u}_1 = 0 \] Now we can compute the inner products: \[ \mathbf{u}_{1}^{H} \mathbf{z} = \mathbf{u}_1^{H}[(4 + 2i)\mathbf{u}_1 + (6 - 5i)\mathbf{u}_2] = (4 + 2i)\mathbf{u}_1^{H}\mathbf{u}_1 + (6 - 5i)\mathbf{u}_1^{H}\mathbf{u}_2 = 4 + 2i \] In a similar way, we can calculate the other inner products: \[ \mathbf{z}^{H} \mathbf{u_1} = [(4 - 2i)\mathbf{u}_1 + (6 + 5i)\mathbf{u}_2]^{H}\mathbf{u}_1 = (4 -2i) \] \[ \mathbf{u}_{2}^{H} \mathbf{z} = \mathbf{u}_1^{H}[(4 + 2i)\mathbf{u}_1 + (6 - 5i)\mathbf{u}_2] = (4 + 2i)\mathbf{u}_2^{H}\mathbf{u}_1 + (6 - 5i)\mathbf{u}_2^{H}\mathbf{u}_2 = 6 - 5i \] \[ \mathbf{z}^{H} \mathbf{u_2} = [(4 - 2i)\mathbf{u}_1 + (6 + 5i)\mathbf{u}_2]^{H}\mathbf{u}_2 = 6 + 5i \] In conclusion, we have found the Hermitian inner products as follows: \[ \mathbf{u}_{1}^{H} \mathbf{z} = 4 + 2i \\ \mathbf{z}^{H} \mathbf{u}_{1} = 4 - 2i \\ \mathbf{u}_{2}^{H} \mathbf{z} = 6 - 5i \\ \mathbf{z}^{H} \mathbf{u}_{2} = 6 + 5i \]

The norm of a vector \(\mathbf{z}\) in an orthonormal basis is given by: \[ \|\mathbf{z}\| = \sqrt{\mathbf{z}^{H}\mathbf{z}} \] So we need to find the value of \(\mathbf{z}^{H}\mathbf{z}\): \[ \mathbf{z}^{H}\mathbf{z} = [(4 - 2i)\mathbf{u}_1 + (6 + 5i)\mathbf{u}_2]^H[(4 + 2i)\mathbf{u}_1 + (6 - 5i)\mathbf{u}_2] \] Using the properties of the Hermitian inner product and the orthonormal basis, we have: \[ \mathbf{z}^{H}\mathbf{z} = (4 - 2i)(4 + 2i) + (6 + 5i)(6 - 5i) = 20 + 50 = 70 \] Now we can find the norm of \(\mathbf{z}\): \[ \|\mathbf{z}\| = \sqrt{\mathbf{z}^{H}\mathbf{z}} = \sqrt{70} \] So the value of the norm of the vector \(\mathbf{z}\) is \(\sqrt{70}\).