Complex inner product spaces

Compute the following complex dot products in $\mathbb{C}^{n}$:

Compute the norm of each of the following vectors in $\mathbb{C}^{n}$:

Suppose $\mathbf{u},\mathbf{v}\in\mathbb{C}^{n}$ and $\mathbf{u}\boldsymbol{\cdot}\mathbf{u}=1$, $\mathbf{v}\boldsymbol{\cdot}\mathbf{v}=2$, and $\mathbf{u}\boldsymbol{\cdot}\mathbf{v}=2i$. Then compute:

1. $(2\mathbf{u}+\mathbf{v})\boldsymbol{\cdot}(\mathbf{u}-\mathbf{v})$.

2. $(i\mathbf{u}-\mathbf{v})\boldsymbol{\cdot}(\mathbf{u}-2i\mathbf{v})$.

Which of the following vectors, if any, are orthogonal?

Consider $\mathbb{C}^{3}$ with the complex dot product. Let

Use the Gram-Schmidt procedure to find an orthogonal basis for $\operatorname{span}\left\{\mathbf{v}_{1},\mathbf{v}_{2}\right\}$. Then find an orthonormal basis.