Solution
= Solution
Define $f:W_1\cap W_2\to U$ by $f(u)=(u,-u)$. It is well-defined by the definition of $U$, and
$$
f(au+bv)=(au+bv,-au-bv)=af(u)+bf(v).
$$
Thus $f$ is a <linear map>. If $f(u)=f(v)$, comparing first components gives $u=v$, proving <injectivity>. Every element of $U$ has the form $(u,-u)=f(u)$, proving <surjectivity>. Hence \b[$f$ is a <linear isomorphism>], with inverse $(u,-u)\mapsto u$, and $\dim U=\dim(W_1\cap W_2)$.