Gram–Schmidt

The Gram-Schmidt methodology is a process that generates an orthonormal basis from a linearly independent basis.

We iteratively apply the procedure from $1$ to $n$ to the original vectors $v_1,v_2,\dots v_n$:

1. Normalise the vector $v_i$ by setting $w_i=\frac{v_i}{\Vert v_i\Vert }$
2. Replace the remaining vectors with new ones by removing the $w_i$ component, i.e. the vector $v_j$ becomes $v_j-⟨v_j,w_j⟩w_j\forall j>i$.

The resulting vectors $w_1,w_2,\dots w_n$ form an orthonormal basis.