7.1.31. Let $\mathbb{R}^n$ be equipped with the inner product $(\mathbf{v}, \mathbf{w}) = \mathbf{v}^T K \mathbf{w}$. Let $L[\mathbf{v}] = \mathbf{r} \mathbf{v}$ where \\
$\mathbf{r}$ is a row vector of size $1 \times n$. (a) Find a formula for the column vector $\mathbf{a}$ such that (7.12)\\
holds for the linear function $L: \mathbb{R}^n \to \mathbb{R}$. (b) Illustrate your result when $\mathbf{r} = (2, -1)$,\\
using (i) the dot product (ii) the weighted inner product $(\mathbf{v}, \mathbf{w}) = 3v_1 w_1 + 2v_2 w_2$,\\
(iii) the inner product induced by $K = \begin{pmatrix} 2 & -1\\ -1 & 3 \end{pmatrix}$.