Orthonormalising a Basis $B = (b_1 ,...,b_4 )$ so that it contains $u_1 = \frac{b_1}{\sqrt{\langle b_1,b_1\rangle }}$

Question

I have just a small question:

I have a scalar product and a Basis $B = (b_1, b_2, b_3, b_4)$. The task is to Orthonormalize the Basis B so that it contains $u_1 = \frac{b_1}{\sqrt{\langle b_1,b_1\rangle }}$.

What is meant be "So that it contains $u_1$" and how do I insure that my answer does? I already know that the Grahm-Schmidt-Algorithm for orthonomalizing is the tool I need for that. I've checked my whole math-script twice and did not find anything that at least mentions a hint...

My idea is that this just means that I have to start the Gram-Schmidt-Algorithm using $u_1$ but I am not quite sure if something else is meant by that. That is why I am asking.

Thank you very much for helping out again.

Answer 1

Yes, $u_1$ is just $b_1$, normed. Start with that.

Then, look for $v_2=b_2+\alpha b_1$ with $\langle b_1,v_2\rangle=0$ then norm it as well, $u_2:=\displaystyle\frac{v_2}{\sqrt{\langle v_2,v_2\rangle}}$, and so on..