Gram-Schmidt algorithm

The Gram-Schmidt algorithm is a recursive algorithm for turning a basis $\mathcal{B}$ for some subspace $V$ into an orthonormal basis $\mathcal{U}$. By theorem, we have the vectors in $\mathcal{U}$:

$${\vec{u}}_i=\frac{{\vec{p}}_i}{\Vert {\vec{u}}_i\Vert }$$

and where the vectors ${\vec{p}}_i$:

$${\vec{p}}_1={\vec{b}}_1$$ $${\vec{p}}_i={\vec{b}}_i-{\text{proj}}_{W_{i-1}}{\vec{b}}_i$$