I have the bilinear form defined by the matrix $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 1 \\ 1 & 1 & 1 \end{pmatrix} $. Now I should find an orthogonal basis. I started by using Gram-Schmidt with the standard basis $e_{ij}$.
$v_1 = e_1=\begin{pmatrix} 1\\0\\0 \end{pmatrix}$
$v_2 = e_2-\frac{<v_1,e_2>}{||v_1||}v_1=\begin{pmatrix} 0\\1\\0 \end{pmatrix}$
$v_3 = e_3-\frac{<v_1,e_3>}{||v_1||}v_1 - \frac{<v_2,e_3>}{||v_2||}v_2=\begin{pmatrix} -1\\-1\\1 \end{pmatrix}$
Now if I test these with my bilinear form, $<v_1,v_2>,<v_1,v_3>$ both yield 0 but $<v_2,v_3>$ yields -1. Looked through the calculations multiple times but can't find the error.
I heard from other students that the order matters when you want to get an orthogonal basis. I thought the order only matters when you want to have a specific basis but Gram-Schmidts always results in an orthogonal basis?