Suppose I have $N$ vectors $v_1, v_2, v_3, ..., v_N$
I already know that I can build an orthonormal basis with the Gram-Schmidt process, but I need to keep the first three fixed, even if they are not orthogonal, i.e. I would like to build vectors in this way:
- $q_1 = \frac{v_1}{\left\lVert{v_1}\right\rVert_2}$
- $q_2 = \frac{v_2}{\left\lVert{v_2}\right\rVert_2}$
- $q_3 = \frac{v_3}{\left\lVert{v_3}\right\rVert_2}$
- $q_4$ orthogonal to $q_1$, $q_2$, $q_3$
- $q_5$ orthogonal to $q_1$, $q_2$, $q_3$, $q_4$
...
- $q_N$ orthogonal to $q_1$, $q_2$, $q_3$, ..., $q_{N-1}$
I've tried using the Gram-Schmidt process, but the fact that $q_1$, $q_2$, $q_3$ are not orthogonal invalidates the Gram-Schmidt procedure.
Any suggestions how I could proceed?