[proofplan] We verify the two axioms of linearity -- additivity and homogeneity -- for the dual map $\alpha^*: W^* \to V^*$ defined by $\alpha^*(\theta) = \theta \circ \alpha$, using the corresponding properties of $\theta$ and $\alpha$. [/proofplan] [step:Verify additivity of $\alpha^*$] For $\theta, \phi \in W^*$ and any $v \in V$: \begin{align*} \alpha^*(\theta + \phi)(v) = (\theta + \phi)(\alpha(v)) = \theta(\alpha(v)) + \phi(\alpha(v)) = \alpha^*(\theta)(v) + \alpha^*(\phi)(v). \end{align*} Since this holds for all $v \in V$: $\alpha^*(\theta + \phi) = \alpha^*(\theta) + \alpha^*(\phi)$. [/step] [step:Verify homogeneity of $\alpha^*$] For $c \in \mathbb{F}$, $\theta \in W^*$, and any $v \in V$: \begin{align*} \alpha^*(c\theta)(v) = (c\theta)(\alpha(v)) = c\,\theta(\alpha(v)) = c\,\alpha^*(\theta)(v). \end{align*} Since this holds for all $v \in V$: $\alpha^*(c\theta) = c\,\alpha^*(\theta)$. [/step]