= Solution
{c}
The <dual space> consists of the <linear functionals> on the original <vector space>, with addition and scalar multiplication defined pointwise:
$$
\boxed{V'=\operatorname{Hom}_F(V,F).}
$$
The <dual map> acts by <function composition> with $T$:
$$
\boxed{T':V'\longrightarrow U',\qquad T'(\lambda)=\lambda\circ T.}
$$
For $u\in U$, its value is $(T'\lambda)(u)=\lambda(Tu)$. The composite is a <linear functional> because both $T$ and $\lambda$ are <linear maps>. Moreover $T'(a\lambda+b\mu)=aT'\lambda+bT'\mu$, so the <dual map> is itself a <linear map>. Its direction is reversed because a measurement on the output of $T$ becomes a measurement on its input.
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