Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof / 6 / b / ii

Define $f:W_1\cap W_2\to U$ by $f(u)=(u,-u)$. It is well-defined by the definition of $U$, andThus $f$ 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 $f$ is a linear isomorphism, with inverse $(u,-u)\mapsto u$, and $\dim U=\dim (W_1\cap W_2)$.