Solution
= Solution
{c}
<Bilinearity> of the real <inner product> gives, for the specified <linear combinations>,
$$
\begin{aligned}
\langle u,v\rangle
&=\left\langle\sum_{i=1}^n x_iv_i,\sum_{j=1}^n y_jv_j\right\rangle\\
&=\sum_{i=1}^n\sum_{j=1}^n x_i y_j\langle v_i,v_j\rangle
=\sum_{i=1}^n\sum_{j=1}^n x_iA_{ij}y_j
=\boxed{x^TAy}.
\end{aligned}
$$
The last equality is the entrywise rule for <matrix multiplication>, with $x^T$ the <transpose> of the column <vector> $x$.