Orthonormality

Definition

Vector space

$\mathbf{v}\perp \mathbf{w}$ and $||\mathbf{v}||=1,||\mathbf{w}||=1$

The two vectors are orthonormal if they are orthogonal unit^[Norm 1] vectors

Facts

For the orthonormal vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$, If $\mathbf{b}={\sum }_{i=1}^n{c}_i{\mathbf{v}}_i$ then, $c_i={\mathbf{v}}_i^{⊺}\mathbf{b}$ (projection onto ${\mathbf{v}}_i$)