Orthonormal bases and projections

An orthonormal basis of a vector space, like $\mathbb{R}^n$ (or $\mathbb{C}^n$), is a basis $B = \{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ where every vector is orthogonal to one another, i.e., 

$$\vec{v}_i \cdot \vec{v}_j = 0 \quad \text{for all } i \neq j,$$

and every vector is a unit vector, i.e., 

$$\vec{v}_i \cdot \vec{v}_i = 1 \quad \text{for all } i.$$

For example, the canonical basis for $\mathbb{R}^4$ (or $\mathbb{C}^4$):

B = \left\{  \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix},  \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix},  ...