Let $U$, $V$ and $W$ be vector spaces over a given field $\mathbb K$. Take two linear maps $f:U\longrightarrow W$ and $g:V\longrightarrow W$ and define:
$$U{}_{f}\times_g V:=\{(u, v)\in U\times V: f(u)=g(v)\}.$$
This is clearly a vector subspace of the direct product space $U\times V$. What can we say about the dual vector space $(U{}_{f}\times_g V)^*$?
I know that $$(U\times V)^*\simeq U^*\times V^*$$ as @Kenny Laus pointed out. For $U{}_f\times_g V$ we can't realize $U$ and $V$ as vector subspaces although we can consider $\mathsf{Ker}(f)$ and $\mathsf{ker}(g)$ as vector subspaces.
Anyone can help me?
Thanks.