Suppose $T: \mathbb{R}^2 \to \mathbb{R}^2$ is defined as $$T(x,y) = \begin{bmatrix}2x-y \\ x+3y \end{bmatrix}$$
Find the adjoint $T^{*}$ of $T$. So we can create a linear functional as follows: Choose $w = (x,y)$. Then $$T_{w}(x,y) = \langle(2x-y,x+3y), (x,y) \rangle$$
So then we need to find the vector $T^{*}w$ such that $$\langle Tv, w \rangle = \langle v, T^{*}w \rangle$$ for all $w \in W$ and $v \in V$.