Solution
= Solution
{c}
A real <inner product> must be symmetric. Here $\beta_a(e_1,e_2)=-2$ and $\beta_a(e_2,e_1)=-a$, forcing $a=2$. For this value,
$$
\beta_2(x,x)=4x_1^2-4x_1x_2+3x_2^2
=4(x_1-x_2/2)^2+2x_2^2.
$$
This is strictly positive for every nonzero <vector>. Bilinearity is already given, so symmetry and <positive definiteness> prove
$$
\boxed{a=2\text{ is the unique admissible value}.}
$$