Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2011 / 2 / c / Solution

Use the inner product with $a=2$. The required first vector $v_1=(1/2,0{)}^T$ already has squared norm $4(1/2{)}^2=1$. Apply the Gram-Schmidt process to $v_1$ followed by $e_2=(0,1{)}^T$. Since $\beta (e_2,v_1)=-1$, its orthogonal remainder is $w=e_2+v_1=(1/2,1{)}^T$. Its squared norm is $\beta (w,w)=1-2+3=2$. Therefore an orthonormal basis containing the specified vector isThe second vector may also be replaced by its negative.