Choice of vectors from basis in Gram-Schmidt process

Question

Say I have a basis for $\mathbb{c}^{2}$ composed of the vectors $(1,1), (4i,2i )$ with complex inner product. When I construct my orthogonal basis using the Gram-Schmidt process how do I make a choice of which of these vectors are $u_{1}$ and $u_{2}$ because this will obviously have an impact when we carry out the inner product and swapping would give different answers.

Answer 1

You can take the vectors in any order you like. You will still get an orthonormal basis from the Gram-Schmidt process (though, in general, a different one).