Orthonormal Basis

An orthonormal basis for a subspace $W$ is an basis for which every vector is Orthonormal, i.e. Orthogonal and Normal If $\{\overrightarrow{u_1},\cdots ,\overrightarrow{u_p}\}$ is an orthonormal basis for $W$:

$$\vec{w}=(\vec{w}\cdot \overrightarrow{u_1})\overrightarrow{u_1}+\cdots +(\vec{w}\cdot \overrightarrow{u_p})\overrightarrow{u_p}$$ $$||\vec{w}||=\sqrt{(\vec{w}\cdot \overrightarrow{u_1}{)}^2+\cdots +(\vec{w}\cdot \overrightarrow{u_p}{)}^2}$$