= Solution
{c}
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 is
$$
\boxed{\left\{\begin{pmatrix}1/2\\0\end{pmatrix},\quad
\frac1{\sqrt2}\begin{pmatrix}1/2\\1\end{pmatrix}\right\}.}
$$
The second vector may also be replaced by its negative.
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